A Momentum Variable Calculation Procedure for Solving Flow at Al1 Speeds

A Momentum Variable Calculation Procedure for Solving Flow at al1 Speeds

Masoud Darbandi

A thesis presented to the University of Waterloo in fulfilment of the thesis tequitement for the degree of Doctor of Philosophy in Mechanical Engineering

Waterloo, Ontario, Canada, 1996

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iii Abstract

The main purpose of this PhD reseatch is to develop a numerical method for solyiog ilc~at all speeds using momentwn eomponent variables instead of the sep- lar velocity ones. The Werent nature of compressible and incompressible goveming equations of fluid flow generaüy elass* the solution techniques into tao main categories of compressible and incompressible methods. However, one extended purpose of this research is to develop an approach which permits incompressible methods to be extended to compressible ones nsing the analogy of flow equations. The pr* posed momentum component variables play a signüicant role for tramferring the individual characteristics of the two formulations to each other in th& adapted forms. In this regard, the two-dimensional Navier-Stokes equations are treated to solve time-dependent laminar flows nom very low speeds, i.e. rdincompressible flow, to supersonic flow. The approach is MyMplieit and employs a contro1- vo1ume-based finiteelement met hod with momentum components, pressure, and temperature as dependent variables. The proper definitions for considering the dual role of momentum components at control surfaces plus the strong connective expressions between the variables on control volume surfaces and the main nodal values remove the possibility of velocity-pressure decouphg and dow the use of a colocated grid arrangement.

The performance of this new formulation is illustrated by solving different types of flow including incompressible and compressible (subsonic to supersonic), viscid and inviscid, and steady and unsteady options. No CFL number limit was encountered for the test models except in supersonic cases. The results demonshate exdent performance of the flow analogy. Acknowledgement

1would like to express my sinaxe appreciation to my supervisor, Professor G.E. Schneider, for his advice and guidance throughoat the course of this work.

1 also wish to express my thaalrs to my former and present office mates (ML R.W. Macdonald and Mr. B. Van Heyst) and the otha members of the Computll- tional Fluid Dynamics group at the University of Waterloo with whom 1 have ben doseiy involved and with whom 1 have had many ioformative discussions. Special thanks are due to Mr. P. Zwart for his special help and &.M. Ashrafizaadeh for his many efforts to maintain the cornputer system for the group and to help the users, including myself.

1 would like to thank the Ministry of Culture and Higher Education of Iran for the postgraduate scholarship awarded to me during the course of this study.

My deepest appreciation is extended to my wife, Fatemeh, who bravely faced the hardest perïod of OUI lives during the first half of this four-year period. The illness of our son tnrned out that fust period into the most bitter one that we had ever seen.

Finally, 1 would like to dedicate this work to my parents, Hossein and Afagh, for their strong support of all of my academic endeavours, and to my children, Adel, Hamed, and Zahra. Contents

1 Introduction 1

1.1 Background ...... 1

1.2 Literature Review ...... 3

1.2.1 Incompressible Flows ...... 5

1.2.2 Compressible Flows ...... 8

1.2.3 Pseudo-Compressible Flows ...... 10

1.2.4 Compressible-Incompressible and All-Speed Flows ...... 13

1.3 Objectives of thig Research ...... 16

1.4 Thesis Outline ...... 18

2 Governing Equations & Research Motivation 20

2.1 Governing Equations ...... 21

2.2 Domain Discretization ...... 25

2.3 Dependent Variables ...... 28

2.4 Research Motivation ...... 29 2.4.1 hcompressible-Compressible Analogy ...... 30

2.4.2 Fluxes versus Primitive Variables ...... 33 2.4.3 Tireating the Nonlinearities ...... 36

2.5 Closnre ...... 39

3 One-Dimensional Investigation and Results 40

3.1 Introduction ...... 40

3.2 One-Dimensional Domain Discret ization ...... 41

3.3 Discretization of Governing Equations ...... 42

3.3.1 Conservation of Mass Equation ...... 43

3.3.2 Conservation of Momentum Equation ...... 44

3.3.3 Conservation of Enagy Equation ...... 45

3.4 Integration Point Operators ...... 46

3.4.1 Momentum Component Variable ...... 47

3.4.2 Other Variables ...... 50

3.5 Pressure-Velocity Decouphg Issue ...... 52

3.5.1 Decoupling in One-Dimensional Formulation ...... 53 3.5.2 A Solution to the Decoupling Problem ...... 55

3.5.3 0th- Possible Solutions ...... 59

3.6 Velocity Component Based Formulation ...... 62

3.7 Applications ...... 65

3.7.1 Incompressible Flow ...... 65

vii 3.7.2 Compressible Flow ......

3.7.3 Cornparison Between Velocity and Momentum Fomdations

3.8 Closure ......

4 Computational Modeling in Two Dimensions

4.1 Introduction ......

4.2 Preliminary Definitions and Descriptions ......

4.3 Discretization of the Governing Equations ......

4.3.1 Conservation of Mass Equation ...... 4.3.2 Conservation of Momentum Equation ...... 4.3.3 Conservation of Energy Equation ......

4.4 Integration Point Equations ......

4.4.1 Momentum Components ......

4.4.2 Pressure ......

4.4.3 Temperature Variable ...... 4.4.4 Other Integration Point Equations ......

4.5 Mass Conserving Connections ......

4.6 Assembly ......

4.7 Boundary Condition Implementation ......

4.8 Closure ...... 5 Two-Dimensional Results 134

5.1 Introduction ...... 134

5.2 Incompressible Flow ...... 136

5.2.1 Driven Cavity Problem ...... 137

5.2.2 Channel Entrance Flow ...... 144

5.3 Pseud~CompressibleFlows ...... 151

5.3.1 Cavity Flow Problem ...... 152

5.3.2 Channel Entrance Flow ...... 159

5.3.3 Converging-Diverging Nozzle Flow ...... 163

5.4 Compressible Flows ...... 166

5.4.1 Compressible Cavity Problem ...... 167

5.4.2 Converging-Diverging Nozzle Problem ...... 171

5.4.3 Flow in a Channel with a Bump ...... 185

5.4.4 Supezsonic Flow in a Channel with a Ramp ...... 194

5.5 Cornparison Between Velocity and Momentum Formulations .... 197

5.6 Closure ...... 201

6 Concluding Remarks 203

6.1 Summary ...... 203

6.2 conclusions ...... 205

6.3 Recommendations for F'uture Research ...... 210 A Element Geometric Relations 223

A.l Introduction ...... 223 A.2 FiniteHement Discretization ...... 224 A.3 Lod-Global Coordinate 'Ransformation ...... 226

B Density Linearization 229

C Linearization of Momentum Convections 231

D Linearization of Momentum Diffusion Terms 236

E Convection in Non-conservative Momentum 238

F Linearization of Energy Tkansient Term

G Linearization of Energy Convection Terms 242

H Streamwise Discretization Approach 244

I Laplacian Operator Discretization 246 List of Tables

"ikeatment of the noalinearities in the mass and momentum equations. 37

Dinerent method of treating e& in mas, consenring equation. . . . 60

Abbreviations and parameter definitions...... 87

Comprehensive cornparison of cavity details for Re =5000 ...... 144 Entrance or developing flow length ...... 149

The hydrodynamic entrance length ...... 150

Comparison of the results of different compressible schemes for cavity flow, Re = 100, grid 19x19 ...... 156

The number of tirne steps to achieve the criterion in compressible entrance flow, M=0.05 ...... 163 Location of the shock in percentage of the bump chord length . . . 191

Comparative study of the convergence histories between the velocity- based and momentum-based formulations...... 200

Mass and linear momentum fluxes for control volume in Figure C.1. 233 List of Figures

2.1 Definitions inside an element ...... 26 2.2 Twwiimensional domain discretization...... 26

2.3 Surface vector demonstration...... 28

2.4 The change of flow properties across a normal shock wave ...... 35

3.1 One-dimensional domain discretization...... 42

3.2 Source and sink distribution in one-dimensional domain...... 66

3.3 The efEect of element pressurc+souree/sink on pressure and velocity distributions ...... 67

3.4 The &ect of control-volume mass-source/sink on pressure and ve-

locity distributions ...... ; ...... 68

3.5 The dect of element mass-soutce/sink on pressure and velocity dis tributions without using convecting momentum equation ...... 69

3.6 The effect of element mas-source/sink on pressure and vdocity distributions ...... 69

3.7 The dect of control-volume pressure-source/sink pressure and velocity distributions without using convecting momentum eqnation. . 70 3.8 The &ect of control-volume pressure-souroe/sink on pressure and velocity distributions......

3.9 Comparing the numerical results of Method 1 and II using control- volume pressure-source/sink in domain...... 3.10 Shock tube problem and its wave pattern......

3.11 Shock tube results for Method I with a =1, G0.4, and 201 nodes.

3.12 Shock tube results for Method 1with a=l, G0.7, and 201 nodes. .

3.13 Shock tube results for Method 1 with a=l,G1.0, and 201 nodes. .

3.14 Shock tube, using Method II with a=l, G0.4, and 201 nodes. . . . 3.15 Shock tube, using Method III with a=2, e0.4, and 201 nodes. . .

3.16 Shock tube results for Method 1 with a=l, G0.4, and 151 nodes. .

3.17 The comparison of mass flux distributions between vdocity-variable and momentum-variable formulations, €=IO-? ......

3.18 The comparison of velocity distributions between velocity-variable and momentum-variable formulations, c= IO-'......

3.19 The comparison of pressure distributions between velocity-variable and moment um-variable formulations, a= 10-3......

3.20 The comparison of density distributions behneen velocity-variable and momentum-variable formulations, a= 10-'......

3.21 A rough comparison between isothermal and non-isothermal exact solutions in the shock tube problem......

3.22 Cornparison of pressure distributions between velocity-variable and momentum-variable formulations, c-10-'...... 3.23 Cornparison of velocity distributions bahreen velwty-variable and moment-variable formulations. S=~O-~...... 83

4.1 A schematic influence of uptaind point ...... 105

4.2 Element nomenclature and velocity upwinding...... 119 4.3 Boundary control volume and boundary condition implementation. 128

Cavity with a 31x31 grid and Re =IO00 ...... 139

Mes h refinement study on centerline velocities ...... 139

Cavity with a 51x51 grid and Re =5000 ...... 140 Mesh refinement study on centerline velocities ...... 140

Cavity with a 71x71 grid and Re =7500 ...... 141 Mesh refinement study on centerline velocities ...... 141

Comparative study on the centerline velocities in cavity. Re=3200 . . 142

Cavity with a 51x51 grid and Re =3200 ...... 143 5.9 Developing and developed zones ...... 145 5.10 Entrance flow. Re=200. undeveloped...... 147 5.11 Entrance flow. Re=200. developed...... 147

5.12 Entrance flow. Re=1000. undeveloped...... 147 5.13 Entrance 0ow. Re=2000. undeveloped...... 147

5.14 The vdocity distributions for vertical center grid of the cavity problem. Re=100 ...... 154 5.15 The velocity distributions for horizontal center grid of the cavity problem. Re=lOO ......

5.16 Cornparison of the convergence histories in canty with Re400...

5.17 Comparison of the convergence histories in cavity with Re=1000 . .

5.18 The velocity distributions for vertical center grid of the cavity prob lem. Re=1000 ......

5.19 The velocity distributions for horizontal center grid of the cavity problem. Re=1000 ......

5.20 Centerline velocit y distribu tions for incompressible and compressible flows in entrance region. Re=20 ......

5.21 Comparing the convergence histories for entrance flow. Re=20 ....

5.22 Centerline velocit y dis tribution for incompressible and compressible fiows in entrance region. Re=2000 ......

5.23 Comparing the convergence histones for entrance flow. Re=2000 . .

5.24 Hyperbolic converging-diverging nozzle configuration. AR=2.0. ...

5.25 Density distributions for four cliffixent low inlet Mach numbers...

5.26 Mach distributions for four different low inlet Mach numbers.....

5.27 Comparison of convergence histories for low inlet Mach number %ows in nozzle......

5.28 Compressible cavity. Re=3200. grid 51 x 5 1. M=0.8......

5.29 Compressible cavity. Re=5000. grid 51 x 51. M~0.8...... 5.30 Compressible cavity. Re=7500. grid 71 x 71. Me0.8...... 5.31 Mach and pressure distributions for an isothamal nozzle. Mi= mO.25.

5.32 Mach contour lines in a converging-divaging nozzie with AR=2.035. 5.33 Isothermd Mach and pressure distributions for the nozzle presented in Figure 5.32 with Min e0.3...... 5.34 Density distribution for three difFerent throat Mach numbers..... 5.35 Mach distribution for three diffixent tbroat Mach numbers...... 5.36 Temperature distribution for three different throat Mach numbern . . 5.37 The effect of mesh size on the Mach contour distribution......

5.38 Mesh size efEect on the accuracy of the numerical solution...... 5.39 Mach distribution in nozzle with M-=1.67...... 5.40 Pressure distribution in nozzle with M-=1.67...... 5.41 Mach distribution dong nozzle with M-=2.1...... 5.42 Pressure distribution dong nozzle wit h M-=Z.I...... 5.43 Temperature distribution dong nozzle with M-=2.1...... 5.44 Mach distribution for fdy supersonic nozzle. M-=2.2...... 5.45 Pressure distribution for fdy supersonic nozzle. M-=2.2. .... 5.46 Isobar lines within a fidl supersonic nozzle, M-=2.2...... 5.47 Grid distribution for subsonic and transonic fl.ows in a channel with bump using 65 x 17 grid distribution...... 5.48 Mach contours for the flow in a channel over a bump. M;,=0.5. ..

5.49 Mach distributions on the walls of Channel and comparing with [92] using 89 x33, [33] ushg 60 x20. and [32] using 67x22 non-uniform grids. M,=0.5...... 5.50 Mach distributions on the walls of channe1 with and without using convecting momentmn equation, Min=0.5- ...... 189

5.51 Mach contours for transonic flow in a channel 6tha bump, M;,=0.675.190

5.52 Mach distribution on the walls of Channel and comparing with [92] using 89 x33, [33] using 61)x 20, and [32] nsing 6'1 x22 non-uniform grids,Min=O.675 ...... 190

5.53 Grid distribution for supersonic flow in a channe1 with bnmp. . . . 192

5.54 Mach contours for supersonic flow in a channel with a bump, M.=1.65.193

5.55 Mach distribution on the walls of channel and comparing with [92] using 89 x33, [33] using 60x20, and [32] using 67x22 non-dom grids, Min=1.65...... 193

5.56 Grid distribntion within a channe1 with a ramp...... 195

5.57 Mach contour lines plot for ramp with Mïa=2.5...... 195

5.58 Cornparkg Mach distribution result at height y=0.75m with other results in a duct with a ramp...... 196

A.l Finite-element discretization within a nozzle domain...... 225

A.2 An isolated element...... 225

C. 1 A control volume showing inlet and outlet mass flows on the x faces. 232

1.1 Laplacian length scale...... 247 Nomenclature

coefficients matrix of variables in element conservation equations right-hand-side vector in element conservation equations coefficient matrix of ip variables in dement conservation eqtiations coefficient matrir of nodal variables in integration point equations vector of hoan values ia integration point equations coefficient matrix of ip variables in integzation point equations Courant number, Eq. (3.30) influence co&cient matrix specific heat at constant pressure spedc heat at constant volume coefficients of elemental stifbess matrix, Eq.(4.167) right-hand-side coeffiuents in elemental stfiess matrir, Eq.(4.167) outward nomal vector to surface, Figure 2.3 total energy per unit mass at node and integration point x-moment- (mass flux) component at node and integration point y-moment- (mass flux) component at node and integration point enthalpy per unit mas distance between two plates, Figure 5.9 unit vector of global coordinate system Jacobian of transformation thmal conductivity constants or optional coefficients diffusion length scale, Eq.0.4) the hydrodynamic entrance lengt h, Eq. (5.3) (up & down)stream lengths, Figure 4.2 Mach number mass flow finite &ment shape functions the total number of grid nodes in solution domain pressure at nodes and integration points, respectively Peclet number, Eq.(3.29) heat flux components gas constant moment um force Reynolds number volumetric source mass source term x-momentum and y-momentum source terms energy source term control-volume surface area, Table 4.1 temperature at node and integration point x-velocity at node and integration point y-velocity at node and integration point velocity vector, ui + vj' global Cartesian coordinates the entrance length, Figure 5.9 Greek weighting co&cient, see Eq. (3.52) the dope of the tangent line at the boundary, Figure 4.3 rfficient &sion coefficient ratio of specific heats time step convergence criterion vor ticity dissipation term t ime viscosity micro seconds local non-orthogonal coordinate system, Figure A.2 density at node and integration point stress tensor components normal and tangentid surface stresses scalar at integration point and node stream function conserve quantity vector at node and integration point, Eq.(2.19)

Miscellaneous convection fluxes, Eq.(2.20) dinnsion flues, Eq. (2.21) the area or the volume per unit depth of control volume Subscripts indicates the downstream values, Figure 4.2 integration point, Figure 2.2 integration points, see Fig.3.1 nodal points, see Fig.3.1 normal tangent id indicates the upstream values, see Figure 4.2 Superscripts previous time step value

Accents time rate of change convecting parameters lagged from previous inner iteration

Acronyms AR aspect ratio or A - bip boundary integration points, Figure 4.3 IEC irrotational entry condition 1P integration points, Figure 2.1 NO1 number of iterations RMS Root-Mean-Square, Eq. (5.1) scv sub-control-volume, Figure 2.1 Si,SSi control-volume edge in xy plane, Figure 2.1 and Table 4.1 UEC domentry condition Chapter 1

Introduction

In this chapter, we are concerned with the general position and spedic situation of this research, the purpose of this research, and its contributions. In this regard, a preliminary background is introduced in Section 1.1. It is followed by a literature review which presents the previous related works in Section 1.2. The objectives of this research are presented and discussed in Section 1.3. The outline of the thesis is presented in Section 1.4, where the progress of the subject during this investigation is discussed.

1.1 Background

This research is concenied with the branch of fluid dynamics known as Computa- tional Fluid Dylamics, or CFD, i.e., cornputer modeling and solving of fluid flow problems, which plays an important role in aerodynamics, fluid dynamics, and heat tramfer.

Prediction of fluid flow and heat transfer problems has traditionally been ob- CHAPTER 1. INTRODUCTION t ained by experimental investigations and t heoretical calculations. It is apparent that the most reliable information about a physical process is often given by actual measurement. However, full scale equipment that gives more accurate results for experimental methods is expensive and often difficult to construct. The majorïty of fluid flow problems of interest in engineering cannot be solved analyticdy because of the complexity of the governing coupled partial differential equations.

Ln numerical methods, the dinerential equations are modeled by a set of dge- braic equations which must be solved by computer. Fortunately, the development of numerical methods and the availability of large digital computers have enabled practical problems to be solved successfdy. Indeed, CFD has made rapid progress since computers were fmt used as a tool in computational research. The contribution of CFD to the related sciences is remarkably high. CFD has also shown great contribution to experimental science where accurate design of experimental apparatus is necessary. Although research progress in numerical methods has been considerable, there is still room for signÜ;cant progress. Almost all research in this branch is focused on not only eliminating the Limitations and drawbacks of previous work but also on achieving an easy and optimum algorithm which would be capable of solving all real fiuid fiow and heat transfer problems. Accuracy of the results and stability of the method are two out of the many important goals in developing CFD codes.

The behavior of fluid flow fiom very low Mach number to hypersonic flow is not the same. Various methods have been presented for solving cliffixent flows and Mach number ranges. Most of these methods have not been successful for 0th- ranges. Many algorithms have ben developed for solving the Euler and Navier-S t okes equations. Also, difFereat cornputational met ho& have been developed through the years to deai with compressible and incompressible flows because CHAPTER 1. INTRODUCTION 3 of the dinaences in the nature of these two types of flow. kntly, maay works have focused on algorithms capable of solving both compressible and incompressible flows. Most of these methods employ primitive variables (velouties, pressure, and temperature) as the dependent variables.

The main objective of this research is to solve compressible and incompressible laminar flows where the dependent variables are momentum components, pressure, and temperature. Although the basic idea for tuning fiom velocity variables to momentum variables was to explore and identûy advantages while removing previous drawbacks, there are other reasons for this switch which are presented in the following sections.

A brief literature review of related works in this field is presented in the next section.

1.2 Literature Review

The literature in the CFD area is vast and it is not desirable to review it d.In this t section a brief review of the relevant literature is given. All dinerent methods in CFD have advantages and disadvantages. In some circumstances, it is very difncult to deude which outweighs the other.

CFD numerical methods are dassified into different categories depending on the nature of the flow and governing equations. For example, a numerical approach may solve only compressible or incompressible, viscous or inviseid, steady or unsteady, or subsonic or supersonic flows. However, the distinction between two dinerent flow types is sometimes so strong that it does not permit extension from one method to another. This difncuky has ben experienced in extenduig compressible and incorn- CHAPTER 1. INTRODUCTION 4

pressible methods to each other. The important role of density (and pressure) in the fluid governing equations has focused attention towards these two completely distinct branches of CFD. Various methods and techniques have been developed through the years to deal with each of them, but there are significant limitations for each in the range of the applicability of the other. From the mathematical viewpoint, the nature of the compressible and incompressible flow equations is also another issue which dects the deveiopment of compressible methods to solve incompressible flows. The unsteady compressible equations are parabolic-hyperbolic in nature, but the incompressible equations are of elliptic-parabolic type. This in- compatibility between the nature of the equations causes computational difnculties in extending methods developed for one regime to the other. These restrictions and obstacles have prompted an increasing effort in CFD for developing codes capable of solving flows for different options including both compressible and incompressible flows. This is also the main concern in the curent study.

One important and critical issue in developing codes for solving compressible and incompressible flow is the selection of the dependent variable set. There are several different options of dependent variables, but here we consider only the primitive variables. The primitive variables may include eit her velocity or momentum components. The distinction is immaterial for pure incompressible flow, where density is constant. However, using momentum components is very attractive for compressible flows, for several reasons. First of al, this formulation may permit existing solution methods for incompressible flows to be extended to cover the entire flow speed range. Secondly, the need to linearize the terms of the governing equations which include momentum variables is removed. For example, the conservative form of the continuity equation is preserved and it no longer needs to be linearized. Finally, mass flux is a constant parameter passing through a shock wave CHAPTER 1. INTRODUCTION 5 whde velocity undergoes large changes. Using momentnm components may result in fewer oscillation around certain discontinuities in the flow. These advantages will be discussed hirther in Section 2.4. Many of these have encourageci compressible fiow solvers to use moment um components instead of velouty components.

Although our interest in this research is in solving the Navier-Stokes equations, the literature review is not restricted to Navier-Stokes methods. Methods for solving incompressible, compressible, pseudo-compressible, and compressible- incompressible flows are considered in turn.

1.2.1 Incompressible Flows

The primary difnculty in modeling incompressible flows lies in the fact that only gradients of pressure appear in the momentum equation and pressure does not explicitly show up in the continuity equation, although it is the continuity constraint that is used to determine the pressure. This diflicdty may lead to decoupiing in the velocity and pressure fields which creates non-physical solutions, Pat ah[l] . This difficulty has led to methods like stream function-vorticity formulations which eiim- inate pressure from the goveming equations. Although stream function-vorticity formulations have been successful for predicting twwiimensional incompressible flow, they are diffcult to extend to three-dimensional flows. Alternatively, the considerable advantages of;the primitive variable formulation have attracted more investigators toward developing incompressible numerical methods. The staggered grid arrangement is a weii-known technique for treating the velocity-pressure decoupling in the primitive variable formulation of finite diffaence methods. In this technique, pressure and velocity variables are treated on two separate grids [Il. The Marker and Cd(MAC) method of Harlow and WelJi [2] could be named as one of CHAPTER 1. INTRODUCTION 6 the successfd pioneering works which uses primitive variables in a staggered grid arrangement.

The segregation of variables has been a well-liked technique to solve for the primitive variables implicitly. Generdy speaking, segregated methods convert the indirect information in the continaity equation into a direct algorithm for the cal- cdation of pressure. This means that they determine velocities trom the solution of momentum conservation equations based on the best possible estimate of the pressure field. Then, pressure is determined from the solution of one or turo Poisson-lüre equations. This process rnust be repeated for updating the velocity and pressure fields. The segregated solutbn can satisfy both mass and momentum conservation equations if the correcting pressures and velocities vanish. Rait hby and Schneider [3], and Patankar [4] have developed methods based on the segregated approach. Using staggered grids in finite-difierence segregated met hods guarantees the coupling of the velocity and pressure fields [3, 41. There are altemative methods which do not use a segregated approach. Zedan and Schneider [5] employed a Sirnultane- ous Variable Approach (SVA), that uses the strong couplhg between variables and solves aIl dependent variables simultaneously. In theh method, the equation for pressure is obtained by substituting the momentum conservative equation, without approximation for t the correspondhg velocity, into the mass conservation equation.

Since the innovation of control-volume methods in CFD, there have been considerable efforts to use these in solving fluid flow problems including incompressible ones. The main advantage of these schemes is the conservation of the consenrative quantities in each Wte control volume. Most of the control-volume-based met h- ods return to the original work of Patankar and Spalding [6] which is based on a pressure correction technique. They employed a semi-implicit segregated scheme, the Semi-Implicit Method for Pressuse-Linked Equations (SIMPLE),nhich requires CHAPTER 1. INTRODUCTION 7 a heavy under-relaxation for the pressure correction to ensure convergence of the solution. A number of improved Mnants of the original SIMPLE algorithm induding SIMPLER were later developed for solving incompressible flows [4]. The SIMPLEbased methods generally use staggered grid arrangement.

Contrary to a staggered grid arrangement in control-volume methods, it is the colocated grid arrangement which needs speual treatment for the coupling of ve locity and pressure. Colocated grid arrangements are mainly non-segregated. The main idea in a colocated arrangement is to consider the role of pressure as an active parameter in the continuity equation to remedy the decoupling. There are different approaches for removing the checkerboard problem in a colocated grid. Baliga and Pat ankar [7] used unequal order veloci ty-pressure interpolation to remove the decoupling problem in t heir control-volume-based finite-element met hod. Unequal-order means, pressure is computed at much fewer grid points than velocity. Prakash and Patankar [SI also developed a non-segregated approach using a control-volume-based finiteelement method with equal-order velocity-pressure interpolation. Rhie and Chow [9] proposed a new technique for removing the decoupling of velocity and pressure in their colocated grid approach for solving incompressible flows. Their technique includes a new method for treating the convected terms at control volume surfaces. These terms are interpolated between main grid points. Schneider and Raw [IO, Il] used a colocated grid approach in th& control- volume-based fiaite-element method which considers the physical auence aspects of the flow in integration point equations. Later, Schneider and Karimian [12] showed that this derived formulation cannot parantee the coupling of velocity and pressure under cert ab circums t ances. Consequently, they proposed a second integration point velocity variable in order to remove the deficiencies of the previous formulation. With this remedy, a strong coupling between pressure and velocity CHAPTER 1. INTRODUCTION 8

was obtained and the checkerboard problem was totally removed. Darbandi and Schneider [13] have also investigated the checkerboard problem in their pressure based momentum-component procedure. They remove the decoupling problem by introducing a second momentum-component variable at the control volume surface in th& colocated grid arrangement.

Peric, Kessler, and Scheuerer [14] present a detailed comparison of two finite- volume solution methods for two-dimensional fluid flows, one with a staggered and the other with a colocated numerical grid. They show that the colocated scheme generally represents more advantages.

1.2.2 Compressible Flows

Compressible flows can be mainly divided into steady and unsteady methods. Most of the steady methods use the unsteady governing equations to integrate over tirne to reach steady flow conditions. Absolute steady methods are mainly space marching methods. The space marching method is used for solving equations which are parabolic in at least one spatial direction, the marching direction. ALishahi and Darbandi [15]solved supersonic flow for wing-body configuration problem by marching in centerline direction and using discrete zona1 approach.

Uns teady methods are divided into explicit and implicit methods. Explicit schemes are subject to one or more stability restrictions on the temporal and spatial step sizes. These restrictions are asually given in terms of a Courant-Friedrich- Lewy (CFL) and viscous stability condition, which limits the time step. A nnmber of early methods such as PaIumbo and Rubin (161 implemented a twestep Lm- Wendroff scheme which advanced the solutions through tirne explicitly. MacCor- mack [17]fotwarded an important progress in explicit methods by introducing a CHAPTER 1. INTRODUCTION 9 predictor-corrector finite-diffetence algorit hm to solve Euler equations. The indu- sion of artificial Vi~cosityin this algorithm has enabled it to solve flows with shock. Modifications have been done on this explicit Euler algorithm, which severly suffers hom the stiffness of the discrete Navier-S tokes approximation, to solve for viscous flow, e.g. MacCormack [la]. The method of Ref. [18] consists of two steps. The first step is an explicit predictor-corrector finite difference stage and yields second order accuracy in time and space. The second step is an impiicit stage. The second step removes the sever stability limitation of the fust explicit method step.

Many advantages of implicit formulations have generdy shifted interests toward these schemes. There are several important Mplicit schemes that use a finite- difference formulation. Beam and Watming [19,201 treated the conservation form of the governing equations by an AD1 based approximate factorization formulation to produce a block tridiagonal linear system of equations. Density, m^rnPnti= components, and total energy are dependent variables in th& method. Shamroth, McDonald, and Briley [21] changed the momentum components to veloeity ones. Briley and McDonald [22] presented anot her implici t approximat e fact orization method for solving the Navier-S tokes equations. The addition of artifid diffusion is required for stability of their method and to catch shock waves. Indeed, the handling of boundary conditions becomes more severe when approximate factorization is used to break a multidimensional problem into a set of one-dimensional problems. Since the innovation of finite-element methods, there have been parallel works for solving compressible flows using finite-element schemes. For example, Baker and Soliman [23] presented an implicit hite-element algorithm to solve compressible flows.

Besides the progress of control-volume methods for incompressible flows, there have been many attempts to develop SIMPLEbased incompressible methods into CHAPTER 1. INTRODUCTION 10 compressible one. One method was developed by Issa and Lockwood [24] who used an approximate form of the momentum equation to relate velocity corrections to pressure corrections in an staggered grid arrangement. Expressing continuity in terms of density and velocity corrections, these relations can then be used to determine an equation for pressure correction. Han [25] has also tried to extend the SIMPLE procedure for compressible flow calculations. However ,det ails of flow discontinuities were not cap tured well due to excessive numerical smearing in Refs. [24, 251. More related works are presented in Section 1.2.4.

Most of the schemes for compressible flow use density as a primary dependent variable and extract pressure fkom an equation of state, e.g. [18]. Since the role of density at low Mach numbers is very small, this approach cannot be used for incompressible flows as it is discussed in Section 1.2.3. There are also many other factors for each compressible method that prevent the use of these algorithms for low Mach numbers. Methods that use pressure as a primary dependent variable do not have the difficulties of the density-based methods because the change of pressure is always finite, irrespective of the flow Mach number. Therefore, it is possible to modXy pressure-based methods to cover the entire spectrnm of Mach numbers. Rhie [26] has presented a pressure-based segregated method for solving Navier-Stokes equations in compressible flows. This method is an extension of his previous incompressible methods [9]. He uses a multi-step pressure correction procedure, wit h implicit treatment of density, to correct the pressure field.

1.2.3 Pseudo-Compressible Flows

There are various design applications, such as automobile, ship, and turbomachinery design which typicdy solve very low speed flows using compressible algorithms. CHAPTER 1. INTRODUCTION 11

Moreover, compressible algorithms are also important in the field of heat transfer where density has significant changes. As said in the previous sub-sections, the difference in nature between incompressible and compressible flows has spawned various computational schemes to be developed to treat these two types of flow. In the case of compressible flows, methods have ben developeci that use density as a primary variable. Such methods are known as density-based methods [18,20]. Con- trary to density-based methods, there are pressure-based ones which use pressure as a primary variable ins tead of density in the incompressible flow cases [IO, 26,27,28].

If it is realized that the incompressible goveming equations are derived from compressible ones, it is reasonable to recognize slightly compressible flow as being constructed fiom the incompressible one. Van Dyke [29] states that slightly compressible flow is a reguiar perturbation of incompressible flow. This promotes the idea of solving low speed flows using either compressible or incompressible algorithms. There are works which extend the original transonic flow solvers to low Mach nnmber applications [30, 311. On the other hand, there are works which are the extension of incompressible schemes and solve for compressible flows [28,32,33].

The simult aneous solution of the governing equations in compressible met h- ods enhances the stability compared with the segregated approaches of the incompressible techniques, Merk et al [34]. This has encouraged many incompressible investigators toward using compressible algorithms in applications [35]. However this intention in switching encounters some major drawbacks. The speed of sound approaches idmity in the incompressible limit ; implementing compressible codes for simulating incompressible flows is not computationaliy efficient. The hyperbolic tirnedependent Navier-Stokes equations become stiff at low Mach numbers because of the great clifference between the largest and the srnaIlest magnitudes of the system eigenvalues, Feng and Merkle 1361. Since the eigendues are the CHAPTER 1. INTRODUCTION 12

speeds of the waves carrying information, one class of information related to the smallest one would be slowly transported within the region, while the the step is limited by the speed of the fas test traveling wave. Nevertheless, the number of tirne steps needed to reach the steady-state solution wili approach infinity for an incompressible algorithm. Hence, convergence to a steady-st ate solution is nsu- ally slow and for time-dependent solutions the permitteci time step becornes very smd. This is why most compressible dgorithms become either very inefficient or inaccurate at low Mach number speeds. Briley et al [37] rescale the equations to improve the convergence, however, the performance and the accuracy of their time-dependent compressible schemes are inadaquate at low Mach numbers , Say M,

There are methods to overcome the difficulty of ushg a compressible scheme to solve for incompressible flows. One method is preconditioning which modifies the time term. This can be made to appear as a new matrix multiplying the theterm in the vector form of the system of equations, Pletcher and Chen [38]. For steady problems, preconditioning is achieved by altering the physical time-derivative terms in the equations. On the other band, for unsteady problems, an additional pseudo- tirne term of a particular form is added to the equations, which changes the nature of the hyperbolic problem which is being advanced in pseudo tirne. Chorin [39] and Steger and Kutler [40] used artScid compressibility in solving the mass consmm tion equation. They fabricated a hyperbolic tirne-dependent system of equations by adding a time derivative of the pressure term to the continuity equation. Kwak et al [41] developed a code using pseudocompressible methods to cornpute specifi- &y incompressible flows. Choi and Merkle [42] similarly used a smd time step to overcome the difficulties of low Mach number speeds in th& implicit factorization scheme.

Merkie and Choi [43] used asymptotic expansions of the Euler equations in th& perturbation method to solve for low Mach numbenr. They added an artifidal time derivative term to the energy equation. Perturbation schemes become inefficient when the Mach number is not very Iow. Another method for solving low Mach number flows is fluz-vector splitting method which treats the stiff terms not only differently in time but also in space [44].

In addition to the three presented methods which focus on low Mach number Eows, there are compressible-incompressible and dl-speed-flow methods which solve for both incompressible and a wide range of compressible flows [32, 33, 451. However there are few researchers who report the performance of th& methods in solving very low Mach number problems, e.g. Chen and Pletcher [45]. Foliowing the methods for solving both compressible and incompressible methods, Darbandi and Schneider [46] developed an analogy based on momentum component variables which enables existing incompressible methods to be extended to solve for compressible flows. The performance of the resulting method was ülustrated by applying to various test cases fiom very low to transonic Mach numbers.

1.2.4 Compressible-Incompressible and AU-Speed Flows

There are few numerical methods which are applicable to both incompressible and compressible flows. The idea of ~olvingfor all flow speeds with just one algorithm, CHAPTER 1. INTRODUCTION 14 however, is not new. The important question is how to consider the dual role of pressure in compressible and incompressible flows. Gendy speaking, the idea of solving flow at all speeds has been studied using different approaches including finite-elemen t , finit e-difference, and cont rol volume met hods. The smch for an algorithm suitable for aU speeds goes back to the work of Harlow and Amsden [47]. They extended the MAC method for solving time dependent fluid flow problems for all Mach numbers. Their method s&s from a stability restriction and its scope of applicability is limited. Zienkiewicz, Szmelter, and Peraire [48] presented a semi- implicit algorithm for the calculation of both compressible and incompressible flows using a finite-element approach. Zienkiewicz and Wu [49] later developed this finite- element algorithm into a general explkit and semi-explicit method. Howevr, th& work is restricted to relatively low supersonic Mach numbers because the use of the non-conservative form of the equations may lead to inaccurate shock prediction. Hauke and Hughes [31] have similady worked in hiteslement context to develop their original compressible method to solve for incompressible lixnits.

The advantages of control-volume-based methods have encouraged recent work using this approach. Most of the control-volume-based niethods for flow at all speeds return to the incompressible work of Patankar and Spalding (61 which is based on a pressure correction technique. Van Doormal, Raithby, and McDonald [50] have show that a pressure-based method can be extended to indude compressible flows. They used modified versions of the SIMPLE code with a staggered grid arrangement. Karki and Patankar [32] present ed another control-vohmebased finite-difference method which is based on the compressible form of the SIMPLER algorithm. The steady-state form of the Navier-Stokes equations was solved in a st aggered grid arrangement wi th generaiized non-ort hogonal coordinates. Their method suffers fiom sensitivity to grid smoothness which is due to the presence of CHAPTER 1. INTRODUCTION 15 the cwatnre terms in the equations. Demirdzic, Lilek, and Peric [51] employed Cartesian based vectors instead of locally fixed based vectors of Karki et al [32], and removed the sensitivity to grid smoothness. Shyy, Chen, and Sun [52] have ab developed a similar procedure for flow at ail speeds using a multigrid algorithm. Lien and Leschziner [53] have induded the turbulent aects in their control-volume based method for solving compressible and incompressible flows.

Chen and Pletcher [45] presented a colocated pressure-based technique for solving the time-dependent Navier-Stokes equations applicable to low Mach numbers. They found that smoothing was not needed to control oscillations in pressure for subsonic flows despite the use of central differences in their finite diffaence ap proach. Raw, Galpin, and Raithby [54] have presented a colocated control-volume method to solve compressible and incompressible flow fields. Special integration point equations were derived at control volume surfaces to connect them to neighboring nodal valaes. Since the pressure of other nodal points also appear in their formulation they change them to lagged values. Karimian and Schneider [33] used the approach of Ref. [50] and developed the incompressible work of Ref. [IO] to solve for compressible flows.

Most of the all-speed met hods are extensions of incompressible methods to corn- pressible flows, e.g. [32, 33,50,52]. Hence, they apply the incompressible primitive variables, i.e., velocity components, as the dependent variables for extending their work to compressible flows. The main objective of this work is to use the momentnm components as the unknown variables. The idea of using momentum component variables instead of velocity variables was introduced by Darbandi and Schneider [13, 5Y. They also examine the performance of their momentum-component formulation in the context of a flow analogy for solving flow at all speeds [56]. Their implicit scheme is based on a colocated grid arrangement. CHAPTER 1. INTRODUCTION

1.3 Objectives of t his Research

To derive numerical solutions for the nodal values of the dependent variables, it is necessary to develop algebraic relations which approximate the governing merential conservation equations. While no single method yet cont ains aIl desirable features while being void of disadvantages, the following attributes and features are sought in the present formulation:

1. Contd Volume Basis Since the fluid flow governing equations are intrinsically conservative it is

preferable t O use numerical met hods t hat ret ain t his property. Control- volume-based approaches have two major advantages. Firstly, they allow exact numerical conservation of the conserved quantities in each finite control volume. This means t hat mass, momentum, and energy are exactly conserved over any number of control volumes and consequently over the entire fluid flow domain. Secondly, t hey provide a physically meaningful intezpretation of the various terms such as fluxes and source terms in the discretized form of the governing equations.

2. Finite Element Method The finite-element concept generdy returns calculus and vector field theory to the construction of disuete simulation aigosithms. The most important advantage of the finite element method is its great flexibility for handling highly complex solution domains. In the fmite element method, the variation of dependent variables consists of grid point values and interpolation between them. CHAPTER 1. UVTRODUCTION

Moomentum Components as Dependent Variables This formulation may permit existing solution methods for incompressible flows to be extended to cover the entire flow speed range. The need for linearizing the terms of the goveming equations which include momentum variables is removed. For example, the conservative form of the continuity eqnation is preserved and it no longer needs to be Iliearized. Mass flux does not change throngh shock waves in supersonic flows and this may cause less oscillations around discontinuities in the solution.

Pressure as a Dependent Variable Pressure is selected as a dependent variable in prefixence to density because the pressure changes are finite at al1 flow speeds as opposed to the density changes which become very small at low Mach numbers. Thedore, pressure- based methods can be extended to solve incompressible flows.

Colocution of Dependent Variables The use of a non-staggered grid arrangement may produce a wavy non- physical pressure field while still satisfying the discrete momentum equations. The geometrical simplicity of the colocated grid arrangement is very attractive and it will be signiticant if the cause of the pressure oscillation can be removed. Boundary condition implementation difnculty and excessive book- keeping are two major objections to staggered grid methods. In addition, the velocities that satisty mass do not necessarily conserve momentum in the same control volume. However, the use of one velocity field, instead of two velocity fields which are used in colocated grid methods, is an advantage of st aeeered =id met hod. CHAPTER 1. INTRODUCTION 18

6. mly Implicit Formulation Although explicit methods are relatively simple to set up, t hey need very small thesteps to maintain st ability. Conversely, the stability of the implicit methods can be maintained over much larger time steps. Large time steps may reduce accoracy of the transient solution, however, this is not important if only the steady-state solution is desired. In contrast to segregated methods, dl dependent variables are solved simultaneously in our fully implicit formulation. This reduces the need for tracking the solution of dependent variables through sequential iteration or time step advancement.

1.4 Thesis Outline

A numerical method calculates the values of the dependent variables at a finite number of locations, named grid points, in the calculation domain. To calculate the domain variables on grid points, it is necessary to disaetize the goveming equations. A discretized equation is an algebraic relation that connects the values of the dependent variables for a group of grid points. The development of an algebraic representation of equations includes a number of steps.

First, the differential equations to be modeled and the dependent variables to be used must be determined. This task is accomplished in Chapter 2. In addition, the research motivation is another issue which is discussed in this chapter. Second, the domain must be discretized. Chapter 2 describes the control volume approach and Appendix A presents the finite-element formulations which are used to expand the necessary geometry relationships. Third, the algebraic representation which approximates the dinerential equations must be developed. This duty is elaborated in two chapters. In Chapter 3, a preliminary investigation is done CHAPTER 1. INTRODUCTION 19 in a one-dimensional context in order to avoid unnecessary multi-dimensionality complexities, to discover and study the deficiencies of the formulation, and to solve them with appropriate techniques. For example, the pressure-veiocity decoapling issue is one which is considered in this chapter. The extension to twcdimensional modeling is accomplished in Chapter 4 to farther ilinstrate points and demonstrate applicability to multi-dimensional flows. The developed method is examined for several difkrent test models in Chapter 5. The test rnodels try to cover the entire range of flow speed conditions. The final chapter, Chapter 6, is where the major contributions and conclusions of this work are siimmarized and where the recommendations for future work are presented. Chapter 2

Governing Equations and Research Motivation

The main purpose of this chapter is to introduce the goveming equations, to present the method for discretizing the solution domain, and to provide the research rn* tivation. The general form of the governing equations is introduced in Section 2.1. In Section 2.2, the discretization of the solution domain is briefly explained. The options for dependent variables are presented in Section 2.3, together with a discussion on the roles of pressure and density in compressible and incompressible flows. The motivation for the current research is that choosing momentum variables has a number of conceptual advantages over velocity mmponents; This is discussed further in Section 2.4. CHAPTER 2. GOVEmGEQUATIONS & RESEARCH MOTIVATION 21

2.1 Governing Equations

The difkrential equations governing the conservation of mass, momentum comp* nents, enagy, and 0th- scdars such as mass fraction and turbuience of kinetic energy can be cast into a general form as, Patankar [l],

where 4 is a general dependent variable and s denotes the volumetric source (ot si&) of 9. The two terms inside the parenthesis represent convective and Musive flues respectively. The cases where + represents mas, momatum, and energy are of particnlar interest. Assume a Newtonian fluid, with constant viscosity and conductivity, and which obeys Stokes' law, the twcdmensional Cartesian form of these equations is

+ a(~h)+ B(wh) - UT- + gr) Nuria + vri- qu) 4- + se (2.5) ae 62 8~ 62 8Y where extemal heat generation and body forces have been neglected. These equations are derived in many texts, e.g. [57], and are referred to as the Navier-Stokes equations. The components of the stress tensor are CHAPTER 2. GOVERNING EQUATIONS & RESEARCH MOTIVATION 22

Using Fourier's laws of heat conduction, the components of heat flox in Eq.(2.5) are written as

The braces, {), indicate those terms which vanish in the incompressible Mt. In addition to the above dxerential equations, one adaryequation, the equation of state, is needed

If the fluid is assumed to be a caloridy perfect gas, the equation of state is writ ten

P = PR^

The following relationships also exist for a perfect gas

Although ç,, 4,and R vary slightly with temperature, they are assnmed constant in this study. If the change in potential energy is neglected, the total energy and enthalpy of the fluid per unit mas for a pdect gas are CHAPTER 2. GOVERNING EQUATIONS & RESEARCH MOTIVATION 23

Alternatively, for compressible flows at very low Mach numbers, the pressure and density becorne less dependent on each other and; in the idediaed limit of incompressible flow, they are completely decoupled. For incompressible flow, the equation of state, Eq.(2.10), collapses to

For isothermal flow, this equation is reduced to

p = constant = p, (2.16)

In this case, the transient term in Eq.(2.2) vanishes and the continuity equation becomes

Therefore, for the idealized incompressible flow case the equation of state, Eq. (2. IO), is replaced by Eq.(2.16) with e given by Eq.(2.13) as for the perfect gas.

Vector Form of Governing Equations

Since the two-dimensional discretization of the governing equations is done based on the vector form of Navier-Stokes equations, it is useful to present this form of the equations here. Equations (2.2-2.5) are expressed in vector form as

where the conserved quantity vector is dehed as CHAPTER 2. GOVERNING EQUATIONS k RESEARCH MOTIVATION 24

Using the definition of ent halpy, Eq. (2.14), the convection and ddbion flux vectors respectively are

The components of the stress tensor and the heat flux vectos, in Eq.(2.21), are defined as before by Eqs.(2.62.8) and Eq.(2.9). The source vector is defined

Euler Flow Governing Equations

The Euler equations represent the special case of the Navier-Stokes equations where the dissipative transport phenornena of viscosity, mass diffusion, and thermal conduction are neglected. Considering t his definition and assuming no source t erms, Eq.(2.18) is reduced to

where 3 and 0 are given by Eq.(2.20). CHAPTER 2. GOVERNUVG EQUATIONS & RESEARCH MOTIVATION 25

2.2 Domain Discretization

The first step to solving conservation eqnations is domain discretkation. The current numerical met hod uses a control-volume-based fiteelement discretizations as introduced in Section 1.3. This technique ras applied to the heat conduction problem by Schneider and Zedan [58] and extended to fluid flow problems by Schneider and Raw (10,111. While these works have seledecl quadrilateral elements throughout the domain, there are other works which employ the same approach with other hite-element shapes. For example, Baliga and PatanLat [7] considered triangu- lar elements. The present met hod is control-volume-based because the elements are used to cons tact the calculation domain wit h nonsverlapping control-volumes which fdl the solution domain. Nodes are located at the element corners, and will be the location of ail unknowns. In this study we consider quacldateral elements which consist of four edges and four vertices.

The basic relations and transformations of the fmite element scheme are presented in Appendix A for qaadrilateral elements. It wili be advantageous if the control volumes are appropriately defined fiom the elements. If we imagine bounded domains around nodes which do not ovedap each other and aU together cover whole solution domain the preliminary tool for employing the control-volume part of the method is formed. In this regard, each element is broken up into four sub-elements by €=O and q=O lines, Figure 2.1. The assemblage of aJi sub-elements which touch a node form the requted bounded domain which is called control-volume, Figure 2.2. It consists of eight hesegments. Conservation balances for mas, momentums, and energy are applied for each control-volume. Since each sub-elements is used to dehe control-volumes, we rename it to a sub-control-volume or simply SCV.

Figure 2.1 illustrates how (=O and r)=O lines have divided the element into four CHAPTER 2. GOVERNING EQUATIONS & RESEARCH MOTIVATION 26

Figure 2.1: Definitions inside an element.

Control Volume LX

Figure 2.2: Twdimensional domain discretization. CHAPTER 2. GOVERNZNG EQUATIONS & RESEARCH MOTIVATION 27

SCVs. A conservative flux discretization of the governing equations is accomplished by lochhg over all elements and then over all SCV faces within each element. The flux passing through each SCV face is assembled to the comxponding oum control volumes. The fluxes are estimated by integrating over the SCV surf'. The argument of such integrals is approxhated by the mid-point of each surface. These mid-points are denoted as integration points and labeled by ip. They are illustrated in Figures 2.1 and 2.2 by crosses. As seen, the subsutfa~e8are simply tagged by SS labels.

This grid arrangement is called the colocated grid because all unhiowns of the problem are located at the same points. The geometric simpliuty of the colocated grid arrangement is very attractive and it will be sigaificant if the pressure checkerboard problem can be removed, as discussed in Section 1.2.1.

The integration over sub-surfaces require knowledge of the normal vector to each sub-surface. Assume that the sub-surface is stretched between points a with (z,y), and Q wit h (2, y)b, Figure 2.3. represents an outward normal vector to the segment line Ü6 if we assume conventionai counter-clockwise travel on the control-volume surface. It is writ ten as

a = (AS). ;+ (AS), 3 whese

(AS), = -A2 = -(zs - 2.) (2.25b)

In Chapter 3, the one-dimensional form of the governing equations is studied. Therefore, a one-dimensional domain discretization is needed. A simple dom one-dimensional mesh is shown and described in Figure 3.1 and Section 3.2. CHAPTER 2. GOVERNING EQUATIONS & RESEARCH MOTWATION 28

Figate 2.3: Surface vec tor demonstration. 2.3 Dependent Variables

The governing equations were introduced in Section 2.1. Each of these equations is fkequently associated with one basic variable. This basic variable is explicitly represented by the time derivative part of each equation. Density, vdocity components, and temperature (or enthalpy) are mainly associated with the mas, momentum, and energy equations, respectively.

Most compressible flow methods solve the Navier-S tokes equations for density, velocity, and temperature and derive pressure from the equation of state. However, in the incompressible iimit, the density is no longer an unknown, and it vanishes from the continuity equation. Although some methods still use density as a dependent variable in the incompressible limit, they suffix fiom slow convergence, Ion accuracy, and instability, Volpe [30]. Instead, most methods for incompressible flow use pressure as the dependent variable and consider the pressure field to be set indirectly by the continuity equation. The choiees for dependent variables were compared by Hauke and Hughes 1311 who present a comparative study on the performance of three different set of dependent variables, excluding momentam CHAPTER 2. GOVERMNG EQUATfONS & RESEARCH MOTNATION 29 variable set, to solve compressible flow for incompressible lixnits. They deduced that pressure is a good choice in the incompressible limit.

In the present work, pressure is selected as a dependent variable. The difficulty introduced with choosing pressure as a dependent variable is that checkerboard fields may arise, as discussed in Section 1.2.1. This problem is overcome with two different definitions for momentum integration point equations, which are discussed in Chapters 3 and 4. With this practice, both compressible and incompressible flows can be com~utedusing the same cornputer program and speafying a proper pressure-densi ty relationship.

For simple orthogonal coordinate systems such as Carteaian, as oppose to the non-orthogonal coordinate systems, the appropriate dependent variables in the momentum equations are the velocity components. There are other choices for non-orthogonal coordinate systems. The Cartesian velocity components have been widely used as the dependent variables in all speed solvers, Section 1.2.4. The advantage is that the governing equations are very simple and boundary condi- t ion application is easy. In this researeh, momentum component s (or, equivalently, mass fluxes; i.e., f =pu and g=p) have been selected as dependent variables for the momentum equations. The reason for this is discussed in Section 2.4. For the energy equation either temperature or enthalpy could be chosen as the dependent variable. In the present work, temperature is chosen, as it facilitates boundary condition application.

2.4 Research Motivation

Generally speaking, there are dinetent reasons for research in a branch of study. The research path is always open to problems which are unsolved. The status of CWTER2. GOVERZWVG EQUATIONS & RESEARCH MOTIVATION 30 the research for this type of problems is completely clear. However, the path of research is not ended as soon as the problem h solved. There are always rmm to improve the side faetors of the method and its solution like stability, efficiency, accntacy, and so on. There are generally two types of re~eafchapproach. In the first category, an existing method is developed by adding extra professional treatments. In the second one, a new approach, i.e. a novel work, is proposed and investigated in order to hdthe pros and cons comparing with the other existing approaches.

We categorize the curcent tesearch as a novel work and in this section we take a deeper look at the motivation for the cmnt research. The compressible- incompressible analogy, methods of linearization, and smooth fluxes through abrupt changes in flow parameters are issues which form the foundation for selecting me mentum variables as the dependent variables in this research. These issues are separately addressed in this section.

2.4.1 Incompressible-Compressible Analogy

The concept of analogy in this study is to develop incompressible methods for solving compressible flows by switching the dependent variables. In order to express the concept of analogy, it is necessary to expand either the conservation equations or the control-volume level of these equations. In this section, we are concerned with the continuity and momentum equations. On the other hand, as a start to analogy implementation, we confine the discussion to Eulez flow, which contains the basic physics of much high-speed flow.

As the first step, we study the control-volume level of consemation equations. Recd the steady form of the Euler equations, Eq.(2.23), with zero source term. This equations could be integrated over an arbitrary control volume. At this stage we are not interested in the details of integration but the final form of the controi- volume level of conservation qationo. These for= are derïved in Chapta 3 for a one-dimensional study and in Chapta 4 for twcdimensional study. The integrated equations for an arbitrary quadrilateral control volume nin redt in the foIlowing set of equations:

J e... Jc..

where cs. means integration over control surface and d3 is a vector normal to the surface of control volume. For an incompressible flow, these equations reduce to

where p* = plp. Here, u,v, and p* are considered as the dependent variables. The variables outside of parentheses are due to non-linearities, and need to be lineatized. Alternatively, for a compressible flow, Eqs.(2.26) can also be expressed as

Je.,. J c.8. CHMTER 2. GOVERNlNG BQUATIONS & RESEARCH MOTNATION 32

Here, f,g, and p are considered as the dependent variables. Ifequations, Eq~(2.27) and Eqs.(2.28) are compared, it is seen that they are the same except for their vector of dependent variables. Equations (2.27) were developed for incompressible flows where u, v, and p' were dependent variables. However, Eqs. (2.28) were developed for compressible flows where f, g, and p were dependent variables. Therefore, the analogy suggests that incompressible control-volume methods using u, u, and p' as the dependent variables could be extended to solve compressible flows by switching to f, g, and p as the dependent variables.

This analogy can also be applied to the non-conservative form of the governing equations. This non-conservative form may be selected as either the main govaning equations for solving the flow field or the equation for deriving integration-point operators in control-volume methods; in order to connect integration points to the main grid points. For incompressible flow, the steady form of the Euler equations becomes

Here, u, v, and p* are considered as the dependent variables. Alternatively, for compressible flow, Eq.(2.23) can be written as

af af + BP u-+v- --- Terms a2 ag az

47 &J + ap U- az +v-ay -ay = T-s CHAPTER 2. GOVERMNG EQUATIONS & RESEARCH MOTIVATION 33

Here, f, g, and p are considered as the dependent variables. There are two more terms in the right-hand-side of the momentum equations which are a result of the nodinesr convection terms. These terms vanish in the incompressible limit. Again, comparing Eqs.(2.29) and Eqs.(2.30) shows that they are the same except for t heir dependent variables. Therefore, according to the aaalogy, incompressible goveming equations which are arranged for u, v, and p* as dependent variables could be extended to compressible governïng equations by switching to f, g, and p as the dependent variables.

Therefore, this analogy provides an important argument for selecting momentum components as the dependent variables.

2.4.2 Flues versus Primitive Variables

In the conservation equations, Eqs.(2.2-2.5), it is seen that the la-hand sides involve the divergence of the flux of some physical quantities:

Rom Eq.(2.2) PY mass flux Rom Eq.(2.3) flux of 2-component of momentum Rom Eq.(2.4) pvY flux of y-component of momentum Rom Eq.(2.5) pe? flux of total energy

Thus the conservation equations deal directly with the flux of mas, momentum, and energy rather than just the primitive variables such as p, p, and c. The conservation equations can be cast in a common generic form as

a a In this generic form, all arguments of the 88, x, 5-t- in Eq~~(2.2-2.5) are collected in a, b, and c flux vectors, respectively. All non-differential ter- are CHAPTER 2. GOVERNrrVG EQUATIONS & RESEARCH MOTIVATION 34 mected in d which is dedthe source vector. The term a is called the solution vector for an unsteady problem. Equation 2.31 iô called the stmng conservation fom of the govetning equations, in contrast to the Navier-Stokes equations which is a weak conservation form. Comparing 6th Eq.(2.18), we see that

For an inviseid flow, Eq.(2.31) reduces to Eq.(2.23). For an unsteady Euler 0ow problem, the proper dependent variables are {p, pu, pv ,pe). However, for steady Euler problems, a marching method with marching in one space direction may be chosen where the components of 3 are considered as the dependent variables [59].

This form of governing equations is popdar in CFD. There are reasons behind this popularity. For example, Anderson et al [60] have shown that the conservative form of the Euler equations ailows shock waves to be captured as weak solutions, thereby circumventing the need to apply shodr-fit king techniques. Furt hermore, in flow fields involving shock waves, there are sharp discontinuities in p, p, u, t, etc., across the shock. Experience has shown that the conservation fom of the governing equations is bet ter to be used with shock-capturing methods. The use of the conservation form does not result in unsatisfactory spatial osdations upstream and downstream of the shock wave, and the solution is generdy smooth and stable [61]. This is sketched in Figure 2.4. This plot depicts the flow aaoss a normal shock wave for a number of flow-field parameters and th& combinations. There are sharp discontinuities for. p, p, and u variables passing through a shock wave. If u,~~,P,P?(P+pu2) b discontinuity position

1 PI P

pu, @ + pi')

Figure 2.4: The change of flow properties across a normal shock wave. these variables are chosen as dependent variables the equations would directly see a large discontinuity which in turn would resdt in numerical mors associated with t heir calculation.

On the 0th- hand, the mass flwc, pu, is constant aerogs the shock wave, i.e.,

Hence, if pu is used as a dependent variable, the stability and accuracy of the solution would be increased. The investigation of the x-momentnm equation will result a similar condusion for the p + pu2 term, i.e.,

Again, although p, p, and u have sharp discontinuities aaoss the shock wave, the fiux variable of p + pu2 remains constant across the shock. If this flux variable is selected as the dependent variable, the conservation equations wouid see no error associated with sharp discontinuities in p, p, and u. CHAPTER 2. GOVERNLlVG EQUATlONS & RESEARCH MOTIVATION 36

Since we are lookiag for developing a method capable of solving both steady and unsteady flows, the solution vector {p, pu, pv, pe) is selected as the dependent variables. However, it was shown in Section 2.3 that pressure should be chosen as a dependent variable rather than density for incompressible flows.

Based on these arguments, we conclude that the appropriate choice for the vector of dependent variables is @, pu, pv). Another argument for the same choice is given in the next section.

2-43 Tkeating the Nonlinearities

The general form of the Navier-Stokes equations is non-linear regardless of the choice of dependent variables. In order to solve them using methods for linear equation systems, the non-linear terms must be linearized. There are many dinerent linearization techniques such as lagging the coefficients, simple iterative update of the equations, Newton linearization to iteratively update the coefficients, Newton linearization with coupling, and many more, Anderson et al [62]. Each of these linearizations ha9 advantages and disadvantages, and generally do not guarantee convergence. Thus they may be restricted to special applications.

Generally speaking, linearization introduces some errors, such as iterative errors, in solution domain during the process of solving the non-linear equations. Iteration is used to solve non-linear equations by choosing an initial guess and updating it by solving the linearized equations until the residuals are reduced to a preset level. Iterative errors vanish if residuals are enough small. Besides the method of linearization, the number of linearized terms within an equation is $80 important. In generd, more iinearization wouid cause higher numerical errors. We should choose a method which requires as few linearizations as possible. Moment- Variable 1 Velocity Variable

Yes Ya

Table 2.1: Beatment of the nonlinearities i;i the mas and momentum equations.

We now consider the number of linearizations required if the velocity or me ment- components are chosen as dependent variables. Table 2.1 provides a comprehensive cornparison of important terms in Eqs.(2.2-2.4) for both velocity and momentum variables. It is important to note that the linearization is done for the cont rol-volume level of discretizat ion where the difkrential forms disappear by integrating over the control-volume surfaces and volume. Therefore, the fist column shows the reai appearance of the nonlinear term in the general form of the governing equations and the other two columns show the linearization of the same term in the CHAPTER 2. GOVEWGEQUATIONS & RESEARCH MOTIVATION 38 control volume discretized form. The words of "Yes" and "No" are used to identify whether the term needs or does not need linearization with respec to the selected variables. In this table we see that for the continuity equation, the transient term needs linearization with both formulations, while the 0th- two terms need to be linearized only when using velocity components. The method of linearization couid be dinerent for different flow-fields and solution method. Compressible-incompressible solvers may use a Newton-Raphson linearization which mtains the important role of density for compressible flows but shihs this role to velocity in the incompressible l;mit, i.e.

For the momentum equations, the convection terms need to be linearized for velocity or momentum formulations. However, the transient term needs to be linearized only for the velocity formulation. The diffusion terms must be linearized for the momentum formulations, as discussed in Appendk D.

Here it must be noted that we are not to compare just the number of noalinearities in these two formulations and reach a conclusion. The method of linearization is another issue which surmounts the number of linearizations. For example, the relative weights of dinusion and convection in the momentum equations are not the same. Difhsion terms are always discretized using elliptical schemes. Since the same technique is used for ail dinusion terms, linearization then does not pose a major problem. However, the linearization of mass flux terms in the continuity equation and convection terms in the momentum equations require more improved treatment. This is because each component of the linearized term, e.g. p and u in the pu term, may require special treatment consistent with the physics of the original nonlinear term. Therefore, a suitable treatment of the nonlinearities is mach CHAPTER 2. GOVERNING EQUATIONS & RESEARCH MOTIVATION 39

important than reducing the number of linearizations.

Thetefore, we sec! that the seleetion of momentum components over veiouty components prondes additional simplicity in the linearization of the governing equations, specially, the continuity equation. This is another reason for nsing the mornentum components as the dependent variables.

2.5 Closure

In this chapter, the general conservation equation and severai versions of the Navier- Stokes equations were introduced. This vas followed by a discussion on the selected dependent variables. It was also explained that if a computational scheme is to be valid for both incompressible and compressible flows, pressure should be selected as a dependent variable in prefaence to density. In addition, several reasons were given for selecting momentum components inst ead of velocit y components as dependent variables. An analogy was introduced which enables incompressible methods to be extended to solve compressible flows. The smoothness of the fluxes through the discontinuities in the flow-field was another reason to switch to momentum components. Finally, it was shown that the momentum-variable formulation leads to fewer linearization difficulties and introduces less erroneous linearization. Chapter 3

One-Dimensional Investigation and Results

3.1 Introduction

The development of a numerical method requires several steps. The initial steps are very important in the progress of subsequent stages. In a multi-dimensional method, there is no better way to test the method initidy than for the simple case of one-dimensional flow. In this chapter, we examine the method for one- dimensional investigations or formulations with some special applications. The procedure is started by discretizing the solution domain in Section 3.2. Then in Section 3.3, we introduce the one-dimensional governing equations and write the statement of conservative for them. Next, we derive the one+dimensiond integration point expressions and operators in Section 3.4. These derived expressions are fkst checked for sound physical behaviour for special flow cases. Lata in Section 3.5, they are checked for the velocity-pressure decouphg problem. The modeling of CHAPTER 3. ONEDIMENSIONAL mSTIGATKN AND RESULTS 41 the velocity-based formulation is accomplished in Section 3.6. In Section 3.7, the method is tested for a number of test problems. In this regard, a source and sink combination is put in the one-dimensional flow and the resulting velouty and pressure field distributions are studied. The resdts codkm the decoupling problem under special circumst ances. The one-dimensional test is followed by ehecking the formulation for high speed flows with shocks. In this regard, the shock tube problem is selected and tested for one-dimensionai inviscîd flow and the resdts are compared with the anaiytical solution. Finally, a direct cornparison between the momentum-variable procedure and velocity-variable produre, inciuding their results, is presented.

3.2 One-Dimensional Domain Discret izat ion

The twdimensional discretization of the solution domain has been fdly discussed in Section 2.2 and Appendix A. For a one-dimensional study, the grid distribution is very simple. The number of nodes, integration points, and SCVs in each element are reduced considerably. Figure 3.1 austrates a simple uniform one-dimensional grid distribution. The Y and Z dimensions have unit lengths.

Control volumes are located between the two crosses while elements are located between the solid circles. The notation used to denote relative control volume location is illustrated in this figure. The subscripts E and W are used to denote the nodal quantities associated with the control volume to the east and west of node P. Similady, e and w are east and west surfaces of the control volume centered at point P. Upper case let ters are assouated with quantities at main nodal grid points, while lower case let ters refèr to quantities at integration points. This convention is normaily respected throughout this thesis. Figure 3.1: One-dimensional domain discretization. 3.3 Discretization of Governing Equations

In t his section the algebraic representation for different terms of the mass, momentnm, and energy equations are derived according to control-volume-based methods. These algebraic equations have a nonlinear form. These nonlinearities need to be iinearized properly in order to use solution techniques applicable to linear algebraic equation systems. The one-dimensional conservative fonn of the governing equations for mass, moment um, and energy are respectively derived fiom Eqs.(2.2-2.5) for flow in z direction as

PU) P.

where I' = iP.The adaryequation is the equation of state for compressible flow, Eq. @.Il), and the constant density assumption for incompressible flow, Eq.(2.16). The other necessary parameters, lilre total energy and enthalpy, are calculated fkom Section 2.1 considering the one-dimensionality. CHAPTER 3. 0NEDlMENSIONA.L INVESTIGATION AND RESULTS 43

3.3.1 Conservation of Mass Equation

Integrating the equation of mass, Eq.(3.1), over a eontrol volume and using the divergence theorem will yield

where and / are taken over the volume and the surf" normal to the x-aPs P SP of the control volume, Figure 3.1. The transient term is approximated by a lumped mas approach in our hilly implicit method

where J represents the volume of the control volume. The superscript ''on denotes the old values of the corresponding parameter. The subscript P refers to the nodal point P. Since e is not a major dependent variable it is properly linearized for compressible flovr using Eq.(B.B). The result is ,p = @ + & P - T. Similady, the source term is apprairimated as the source strength evaluated at point P times the volume of control volume

The remaining flux term is simply integrated over the boundary surface of the cont rol volume

f. f. and f, are evaluated at the integration points which are not nodal locations. The expressions which relate them to the main grid nodes are derived in Section CHAPTER 3. ONEDIMENSIONAL INVESTIGATION AND RESULTS 44

3.3.2 Conservation of Momentum Equation

Integrating the momentum consavation equation, Eq. (3.2), over an arbitrary control volume wjll yield

The transient and source terms can be treated as before

The convection term is simply integrated over the control volume surfaces

These nonlinear terms at the control volume surfaces must also be linearized. It is possible to linearize them respect to f variable or both f and u variables. The da tails are given in Appendix C and the result is writ ten here by employing Eq. (C. 14)

The pressure term is mitten as

The last term is the diffusion term which is treated in a similar manner CHAPTER 3. 0NE-DIMENSfONA.L INVESTIGATION AND RESULTS 45

The results do not directly involve the main dependent variables and an appropriate substitution is needed. Appendix D represents différent methods to linearize terms si& to the $ term. Here, the scheme of Eq.(D.P) is used to finearize the dinusion term. mer substituting, the results are written as

The terms in the second bracelet can be evaluated by either using the lagged values of the density or by employing the method of Appendix B and considering them as active terms. As is seen, all terms except the transient and source terms need to be evaluated at the integration points which are not nodai locations. The necessary connections are developed in Section 3.4.

3.3.3 Conservation of Energy Equation

In this research, different methods for treating the one-dimensionai energy equation have ben used because the number of nonlinear terms is much more than the number in the momentum equation and they can be linearized dinerently. The final formulation which is presented in this sub-section presents just one which the final one-dimensional results are based on. Integrating the equation of energy, Eq.(3.3), over an arbitrary control volume nill yield

The transient and source terms can be treated as before. However, the nonlinear transient term, i.e. pe, is linearized with respect to p and e using Eq.(B.6) in Appendix B. This gives pe = pe + ëp - jjë where a simple linearization for e is Using Eq.(B.5) for density and substituting it and Eq.(3.17) into the linearized form of pe d hdyyield

The subscript P for the tmsin braces means that all terms are evdnated at the nodal location P. On the other hand, the convection term in Eq. (3.16), f e, is fLst linearized with respect to f and e using Eq.(B.G). Then one more hearization is done for the definition of e, Eq.(3.17). A combination of these two hearizations will resuit in the following iinearized form for the convection tenn,

where k = ~t + tr2. The viscous term in Eq.(3.16) is treated similady top the viscous term of the momentum equation. The extra u in the viscous tenn of the energy equation is lagged. The conduction term in Eq.(3.16) is approxhated at control volume surfaces by writing a central difference involving its neighboring nodal temperatures. The pressure term is changed to up = u(pRt)= R f t. Then using Eq.(B.G) we obtain

- Lfw]+ 1f.k - f,L] - [(me - (m) (3.20)

As before, all terms except the transient term were derived at integration points. The necessary connecting expressions between integration points and the main grid points are derived in the next section.

3.4 Integration Point Operators

The consemative treatment of mas, momentum, and energy was presented in the last section. However, to make the system of algebraic equations wd-posed, the CHAPTER 3. ONEDIMENSIONAL NVESTIGATION AND RESULTS 47

desived discretized fonas require that the dependent variables at control volume surf' be represented in termg of nodal variables. Thetefore, it is necessary to derive expressions for the major dependent variables of the formulation (i.e., momentum component, pressnre, and temperature variables) at control volume surfaces in terms of nodal values.

3.4.1 Momentum Component Variable

Although the continuity equation is treated as an equation for pressure it does not directly involve pressure in itself. This is critical for incompressible flow conditions. A simple connection of f. and f, in Eq.(3.7) to the neighbouring nodal F's does not directly brings the effect of pressure into the continuity eqnation. Moreover, it ensures the eheckerboard problem WU exist [l]. Various techniques have been adopted to overcome this shortcoming in the continuity equation. Many of them have paid attention to a better modeling of the integration point variables. In this regard, new schemes have tried to bring the correct physical aspects of the flow into the integration point equation. Baliga et al [7, 631, Prakash and Patankar [8], and Prakash [64] are few among many works who have presented profiles that attempt to include the relevant physics of the problem. Consequently, Schneider and Raw (101 employed a different approach that uses the governing equations, themselves, to derive the integration point equations. According to their work, an algebraic approximation to the appropriate differential equation is generated at each integration point which consequently WUindude dl of the physics and relevant couplings for that variable. This method will be employed in this work to derive the necessary integration point equations.

Following the work of Schneider and Raw [IO], momentum integration point CHAPTER 3. ONEDMENSIONAL INVESTIGATION AND RESULTS 48 eqnations are derived by apprhatiag the non-consemative form of the momentum equation, which is derived fiom Eq.(3.2). It is

8f (3.21)

The terms of this equation are treated in a manner that tepresents the correct physical aspects of the flow. If these is a strong flow dong a specinc direction, then significant influences travel only fiom upstream to downstream in this direction. So, the conditions at a particdar point can be afkcted significantly by upstream conditions. The balance between diausion and convection terms depends on th& relative strengths. If convection is large enough then it overwhelms the elliptic efFect of the diffusion term. Considering a correct physicd treatment would reqnire a central difference approximation for the pressure gradient term and an upwind approximation for the convection tem. A central differenchg scheme is also used for the diffusion term which reflects the elliptical influence of parameters in the flow field. The transient term is treated as before by a backward diffaence in time. So, the terms of Eq.(3.21) are discretized as CHAPTER 3. 0NeDIMENSIONA.L INVESTIGATlON AND RESULTS 49

Here, both convection terms are treated in an upwind manner. Appendix C presents a comprehensive discussion on the role of the veiocity components in the momentum's convection terms. These roIes could have a direct &ect on the way that these convection terms are diseretized. However, it is shown that if is treated in a central difference manner it produces poor results, Appendix E. Another form of treatment for Eq.(3.24) is

This scheme will include just je and is directly transferred to the unknown part of the equation on the left hand side. The reason for avoiding this scheme is the possibility of poor stability of the method as is shown in AppendLr E.

Substitution of the diseretized terms into Eq.(3.21) and remangement yieid the following expression for the momentum integration point equation

where P and C are the grid Peclet and Courant numbers, i.e.,

As is seen, there is a strong comection between the momentum integration point vMable and neighbouring nodal dependent variables. The use of this expression in the mass conservation equation, Eq.(3.7), provides a couphg of pressure and momentum-variables and helps to eliminate the need for a staggered grid. Although CHAPTER 3. ONEDIMENSIONAL INVESTIGATION AND RESULTS 50 this procedure reduces the necessity of nsing a staggered grid, it may still reveal the checkerboard problem under special circumstances which are describeci in the next section.

If P is changed nom zero to infinie, the influence of upstream and downstream nodd values is properly seen. This is examined for limiting values of the Peclet nuxnber in steady-state condition

The influence of the momentum nodal values properly changes from that of My elliptic for P + O, to that of fdy parabolic for P -+ oc. This demonstrates a correct behaviour of the derived expression for the momentum component at the integration point. In the next sub-section, the ot her integration point variables are exazuined.

3.4.2 Other Variables

There is no direct equation to be used for deriving an integration point equation for pressure. Schneider and Raw [IO] used the pressure Poisson equation as an explicit partial differential equation to show that pressure is a strongly elliptic variable in incompressible flows. Thus, a linear interpolation is used to determine the integration point pressure which is analogous to conventional procedures insofar as its implementation is concerned. CHAPTER 3. ONE-DIMENSIONAL INV23STIGATION AND RESULTS 51

The temperature integration point equation is obtained by directly discretizing the energy governing equation. The onedimensional energy equation for transient flow couid be written in the following form

The tems of this equation are approrimated very sidar to the apprbatiuns in Eqs.(3.22-3.26). The transient term is discretized by a badsward difference, an upwind difference is considered for the second term on lefbhand-side, and the third term of the left-side is approximated by a centrd clifference. The substitution of t hese approximations into Eq. (3.34) yidds

-0 (3.35) where the lagged dissipation term, e, is treated in the foliowing form:

Thus, the temperature integration point equation could be derived &er some rearrangement in Eq.(3.35), i.e.,

The steady-state form of this equation is reduced to

The unlrnown density at an integration point can be calcdated fiom either the equation of state or the nonlinear form of the continuity equation. The importance of this variable is for connecting the momentum and velocity-variable quantities at integration points. In the first case, the equation of state is linearized with respect CHAPTER 3. ONEDIMENSIONAL INVESTIGATION AND RESULTS 52 to pressure and temperature, Appendk B, and then the pressure and temperature integration point equations are substituted. This form of linearization vas not employed in solving the test cases which are presented in Section 3.7. The second case employs the non-conservative form of the continuit y equation, i.e.,

Using a backward scheme for the transient term and an upwind scheme for the second tenn will tesult in

in which the last term of Eq.(3.39) has been lagged. Note that the role of density integration point equation in moment um-variable formulation is not as critical as it is for velocity-variable procedures which uses density directly in the conservative treatment of the continuity equation.

The remaining variables which are not mentioned here , like velocity, do not appear in the formulation with an active role but passive and lagged from previous iterations. Velocity is obtained by using the u = f definition. Other similslrly lagged variables are properly calculated by th& basic definitions, Section 2.1, using the magnitude of major dependent variables at integration points.

3.5 Pressure-Velocity Decoupling Issue

The absence of pressure in the continuity equation and the use of a central merence for the pressure term in the momentum equation permit the pressure field to accept a zig-zag solution in incompressible flows, Patankar [l].Such a zig-zag field, which is known as the pressure checkerboard problem, is not physical. Any nnmber of CHAPTER 3. ONE-DIMENSIONAL INVESTIGATION AND RESULTS 53

additional solutions can be umstructed by adding a checkerboard pressure field to a smooth pressure field solution.

In this section, the pressure checkerboard problem is investigated for the derived discretized equations of Section 3.3 when the integration point equations of Section 3.4 are ernployed. This study is started by simplifying the formulations for steady incompressible flow and testing it for speeial domain situations.

3S. 1 Decoupling in One-Dimensional Formulation

We have already derived the momentum integration point equation, Eq.(3.28), which illustrates a strong coupling between the pressure and velocity fields. This compensates for the absence of pressure in the continuity equation. There remains a question of whether the velocity-pressure decoupling issue has been eliminated. It is intaesthg to investigate this in our colocated formulation which indudes pressure eff'ects in the discretized form of the continuity equation. Consider the control volume Iocated at point P in Figure 3.1, the control-volume discretized equations of mass and momentum for steady-state, incompressible, Euler flow without any source terms are written

Here, fe and fw are derived from Eq.(3.31) considering incompressible fiow conditions with no source term. They are CHAPTER 3. ONEDWNSIONAL INVESTIGATION AND RESUCTS 54

The value of p* is obtained from Eq. (3.33). A shikinterpolation for P, is p, - F.If these four integration point expressions are sabstituted into Eqs.(3.41) the folIowing results are obtained after some r-angement

If the continuity and the momentum equations are respectively mdtiplied by 9 and :, the consideration of

m = (pu). = (pu), = mass flow rate = constant (3.44) will reduce these equations to

Hem, the pressure term in the continuity equation acts as a source term to correct the momentum field. Schneider and Karimian [12] have shown that a zig-zag pressure field results in a zig-zag velocity field in their simila formulation. Following their work, a pressure field similar to

can result in a zig-zag momentum field like CHAPTER 3. O~D~NSfONALINVESTIGATION AND RESULTS

indicating that only every other node is connected. The non-physicai solution set, &s.(3.46 and 3.47), satisfies Eq43.45). In other words, a non-physicaî solution satisfies the discretized governing equations.

A similar conclusion is obtained by direct examinhg of Eqs.(3.41). If u. = u, and p. = pw, then Eqi~(3.41)become

Now, irrespective of any definition used for f. and f,, substitution of the first equation into the second one gives

Considering the pe =-lPp and =pp:Pw approximations will result

This shows that the possibility of a checkerboard problem still exists in this col* cated grid arrangement even with the strong coupling of pressure and momentum dependent variables. Thus, the appearance of pressure in the continuity equation is not the complete remedy for a colocated grid approach. The next sub-section introduces a remedy to eliminate this checkerboard dSculty.

3.5.2 A Solution to the Decoupling Problem

A general remedy to the pressure-velocity decoupling problem is the employment of a staggered grid scheme [l]. However, the current method has been established on a colocated grid arrangement which is proposed as being an advantage of the method, CHAPTER 3. ONEDIMENSIONAL IEiVESTIGATION AND RESULTS 56

Section 1.3. The dünculty could be overcome by introducing a new set of momentam components at integratioii points. The new set should not ody involve all aspects

and physics of the flow but &O consider the satisfaction of mass which the single integration point equation does not. This proposed new set of vaaiables could be used in conjunction with the continuity equation and wherever conservation of mass is addressed. Karimian and Schneider [33] present an argument to obtain the best new set of velocity components at integration points. It is deduced that the eqiiation which involves both momentum and continuity eqaation mors works rd. Folioï+ng their conclusion, a tweparts equation is presented to derive the second set of momentum components at integration points

(Moment- Equation Error) - ~(ContinuityEquation Error) = O (3.51)

The main idea is to invoke the role of conservation of mass in the integration point equations. Such an equation has the dect of invoking momentum and mass conservation, even though it is indirect. We write a general form of this equation using Eqs.(3.1 and 3.2)

where a is an arbitrary coefficient which determines the degree to which the continuity equation error is involved. The discretization of this equation is exactly similar to what was done for the momentum integration point equation, Eqs.(3.22- 3.26). In this regard, similar terms in the two braces are added and transient terms CHAPTER 3. ONEDIMENSIONAL INVESTIGATfON AND RESULTS 57 are treated like previous treatments for the transient terms. The final result is

This new integration point expression is dif£"rent fiom Eq.(3.28). We name it convecting rnomentum or mas conserving momentum because the conservation of mm is included in it. The hat on this variable, f,distinguishes it fkom the convected one, f, which was preseted before by Eq.(3.28). Comparing Eq.(3.53) with Eq.(3.28) shows that the convecting momentam and convected momentum are the same for a=O. The general convecting expression can be simplified to the special case of steady-state, Euler flow, i.e.,

Differing values of the arbitrary coefficient, a, will change the influence of nodal parameter values. This is shown for two different values of a in incompressible flow

Before closing this section, it is instructive to show that the use of two integration point equations for the momentum components does not permit the checkerboard problem to persist. In this regard, Eqs.(3.48) are umitten in the forms that CHAPTER 3. ONEDZMENSIONAL IlWESTXGATXON AND RESULTS 58

involve bot h convected and convecting momentnms

where fe and f, are substituted from Eqs.(3.42). Similar convecting momentums are derived from Eq. (3.56) considering incompressible flow conditions with no source t erm

By comparing with Eqs.(3.42), it is possible to relate the convected and convecting momentums, Le.,

These expressions are substituted in Eq.(3.57a). Assuming ue=u, will result in

This result can be substit uted in Eq.(3.57b) where p, and p, are approximated by Linear interpolation of the neighboring nodal pressures. The result is

Contrary to Eq.(3.50), here adjacent nodes are connected and thus the checkerboard problem as ne know it does not arise. A similar investigation shows that a zigzag pressure field like Eq.(3.46) caanot result in s zigzag momentum/veloQty field like CHAPTER 3. ONEDIMENSIONAL INVESTIGATION AND RESULTS 59

Eq.(3.47). This new momentum convectàng equation is a remedy for elimination of the checkerboard problem. In the next section, other possible techniques for deriving convecting momentum-variables are described.

3.5.3 Ot her Possible Solutions

Although the discretization of Eq.(3.52) was done similar to what had been done for Eq.(3.21), it is possible to treat the convection terms in dinerent manners to get different formulations. Both the convected equation, Eq.(3.31), and the convecting equation, Eq.(3.54), show that the momentum integration value is related to its upstream nodal momentum value for steady-state Euler flow. However, it can be argued that the integration point values should depend on both upstream and downstream nodal values. To examine this, the uvterm, which is a common term in both the continuity and momentum equations, is discretized in different ways. Once more, Eq.(3.52) is repeated hem while the concern is only on uq terms in those two braces, i.e.,

=O (3.62)

~0rnentÜ.mErrm Con t inui ty Error Considering the physical int erpretat ion of the UVtm , both the continuity and momentum equation parts could result in different disaethg options. Table 3.1 summarizes some of the possibilities. The hst row of the table represents Method 1 which was used in the preceding sub-section to derive je.The resdts of that formulation were presented there. Now, if central diffkrencing is employed to the teim in both braces the method is named Method II. The result of this approach CHAPTER 3. ONEDIMENSIONAL mSTIGATION AND RESULTS 60

II II in Moment- Eqaation 1 in Continuity Eqnation (1

Method III Upwind Diffaence Central Diffaence Method IV Central Diff'ence Upwind Difference

Table 3.1: Different method of treating in mass conserving equation.

for steady-st ate Euler flow is

The influence of neighbouring momentum nodal values is again changed if a is changed. This is examined for incompressible flow

The change of a from 1 to 2 provides for this interpolation to be changed from an absolute upwind to an absolute average of neighbouring nodal momentum values.

Another way for treating the UVterm is a combination of upwind differencing for the momentum part and central differencing for the continuity part, Method III. This form ratifies the convecting prescription for the continuity equation and the convected prescription for the momentum equation. The resdt for steady-state Euler %ow is CHAPTER 3. ONEDIMENSIONAL INVESTIGATION AND RESULTS 61

When a + O, this equation approaches the limiting convected equation, Eq. (3.31). In this case, both convected and convecting momentums are the same. However, for a=2 in incompressible flows, the average of neighbouring nodal momentums results for approximating the integration point equation,

There is a fourth method, Method IV, which uses central differencing for the momentum part and upwind differencing for the continuity part. The upwind mode1 was used to treat the continuity parts in Method 1 where the purpose was to make a consistent discretization for similar differential terms of the equation. Generaliy speaking, the main idea in deriving a convecting equation is to employ upwind differencing for momentum convection terms and central differencing for the continuity terms, however, Method IV is presented as yet another possible option. The result of this case for steady-state Euler flow is

Az (dl - c&sm) (3.68) 2üe(1 - a) Considering a=O reduces this equation to the following form for incompressible flow conditions:

which is an inappropriate expression for approximating the momentum integration point value. However, considering a=2 improves it to an appropriate expression

These different methods were studied in a one-dimensional procedue for which resuits are presented in Section 3.7. CHAPTER 3. ONEDIMENSIONAL INVESTIGATION AND RESULTS 62

3.6 Velocity Component Based Formulation

At this stage, the one-dimensional modeling for the moment um component formulation is complete. This formulation provides the mechanism with which to examine Merent onc+dimensional test problems with regards to its ability to solve such problems. However, the solution of these test problems by itself does not provide any advantages or disadvantages wit h respect to the velocity-variable formulation. Where feasible, a compatison will be made of the tesults of the momentam-variable formulation with those of the velocity-variable formulation. In this respect, we wili be concerned with methods which have been developed for solving compressible and incompressible flows at d-speeds. The work of Karimian and Schneider [33] is the best among the available methods for such flows. They provide a damping mechanism in th& formulation in order to avoid spatial oscillations in the vicinity of discontinuities and shocks. This damping also results in improved convergence. Since the current method is free fiom employing any damping mechanism, a cornparison between the two formulations wili be performed on the bais of there being no damping or density averaging in either method.

In order to pursue this latter idea, the one-dimensional approach of Karimian and Schneider [65], who apply it to the shock tube problem, is formulated in the folîowing. A dearer comparison WUbe obtained if the energy equation can be separated nom the continuity and momentum equstions. In th& way only the flow couphgs are considered and the additional complications of the energy equation and its couplings are avoided. Thus, an isothermal flow is assumed throughout the domain and a constant temperature field is specified via the energy equation. The statement of consetvative for the mass and momentum equationa can be obtained fkom Eqs. (3.1 and 3.2) assoming zero diffusion and source terms. The final form of CHAPTER 3. 0lvFDIMENSIONA.L INVESTIGATION AND RESULTS 63

these statements ean be directly written fkom Sections 3.3.1 and 3.3.2 as

The transient term in the continuity equation and the pressure term in the momentum equation are treated as in Sections 3.3.1 and 3.3.2, respectively. The convection term in the moment um equation is simply linearized respect to the velocity-variable, i.e. pu = (F)u.Aowever, the mass flux terms in the continuity equation require special treatment. They are treated by using a Newton-Raphson linearization, Eq.(B.G). This treatment, which was introduced in Section 2.4.3, allows the strong importance of density in highly compressible flows to shift to that of velocity in the incompressible limit. The mass flux is given by

The non-linear density in the second term on the right-hand-side is determined by

The non-linear transient term in the momentum equation is fi~sttreated by a Newton-Raphson linearization, i.e. eU = GU + De - p. Then, the non-linear density is treated by using Eq. (B.5).

In the next step, the convected velocity at the integration point is derived from the non-conservative form of the moment um equation, i.e., CHAPTER 3. ONEDIMENSION' INVESTIGATION AND RESULTS 64

Wnting a backward diffetence in tirne, an upwind difference approximation for the convection term, and a central difference approximation for the pressure term and rearranging the resulting terms give the following expression

This integration point veloaty is the convected one which is substituted into the convection term of the momentum equation. SimiIarly, the convecting velocity at the integration point is derived fiom the following equation

where the term in the fust parenthesis is modeled using central differencing for the spatial derivatives and backward Merencing in time for the transient term. The following result is obtained &er some rearrangement

L 'L Ue = (UP UB) (PP- 9)+ TERMS (3.78) 1 +2@ + + Peu& + 2C) where

Ge üec u: TERMS = (Pe - PZ) + pe(l + 2@) (ës -@Pl + 1 +2c (3.79) AU+ 2@) This convecting velocity at the integration point is substituted into the mass flux terms of the continuity equation, Eq.(3.73).

Now, the modeling of velocity-variable formulation is now complete. HOW- ever, Karimian and Schneider [65] improve this formulation by treating TERMS in Eq.(3.78) and the second term on right-hand side of Eq.(3.74) using an absohrte hamonic interpolation scheme. This treatment damps oscillations in the vicinity of shock waves and discontinuities in the numerical solution. CHAPTER 3. ONEDIMENSIONAL INVESTIGATION AND RESULTS 65

3.7 Applications

To check the accnracy of our nudealsolutions, a cornparison will be made with an exact dytical solution for incompressible and compressible Euler flows. Although our interest in this research is in solving the complete Navier-Stokes equations, the Euler equations are of importance in many flows and do exhibit the strong coupling between velocity, pressure, and density. They thaefore provide a good preliminary tool to evaluate the performance of a NavkStokes solver. In this section, source and sink test cases are hst applied to incompressible flows and the resdts are eramined. Subsequently, the shock tube problem is examined to examine the compressible part of the method. Findy, a direct comparison between the results of momentum-based and velocity-based formulations is provided.

3.7.1 Incompressible Flow

The main purpose here is to examine the checkerboard problem in a flow field when a sudden positive or negative change in mass or pressure is imposed. Therefore, the source terms in the continuity and momentum equations have non-zero values. The test case is a one-dimensional, steady-state, Euler flow through a constant area channel with unit length and with mass/pressure-sowce/sinlt inside it. There are twenty-one dormnode distributions. The source and sink divide the domain into tbree equal parts, Figure 3.2. For the sake of space, some part of domain has not been shown in this figure. The mass is specified at the upstream boundary and pressure is defined at the downstream boundary of the domain for boundary condition implementation. Also, mass and pressure are nondimensionalized with respect to inlet momentum component, Fint and idet dynamic pressure, ( f eU&), respectively. Two diBetent possible arrangements are considered for each source/sink. They are CHAPTER 3. ONEDIMENSIONAL WSTIGATIONAlVD RESULTS 66

Ekmcnt SourMink Disrribution Conml Volume Source/Sink Distribution

Figare 3.2: Source and si& distribution in ont+dimensional domain. dehed by either two neighbonring nodes, named an element eource/sinl, or by two neighbouring integration points, named a control volume source/sink, Figure 3.2. If the source is distnbuted within an element, it is equaliy split between the two control volumes of t hat element.

The discretized equations of continuity and momentum have analytical solution for mass and pressure variables when element pressure-source/sink is located in the domain. For this case, the results of the code are codkmed by the analytical solution, Figure 3.3. This analytical solution cannot be obtained for other types of source and sink distributions. For the control-volume source/sink, there is some oscillations around the source and sink in the numerical results. These local oscillations around the source and sink do not reflect the checkerboard problem. For example, Figure 3.4 shows the results of code for locating a control-volume mass-source/sink in the domain. The numerical result does not fit the required distribution in the domain. In these distributions, the migration fiom before to after a source happens within more than one discrete distance between two integration points or nodes. Figures 3.5 and 3.6 show the results of the code for loc&ing an element mass-source/sink in the domain. One applies the mass conserving ap proach and the 0th- does not. If the mass conserving equation is not applied in the CHAPTER 3. ONEDIIMENSIONAL INVESTIGATION AND RESULTS 67

-Anaiydcai and present solution for node O Anaiytical and premnt mlutloci for lg. x

'8 Ô.1 02 6.3 0.4 Ô$ 0.6 , 0.7 0;8 0.9 Node and 1.p distribution

Figure 3.3: The dect of element pressure-source/sink on pressure and velocity distributions. continuity equation instead of convected one, the pressure checkerboard problem appears, Figure 3.5, while the use of a convecting equation does no t aiiow the growth of non-physical oscillations in the domain, Figure 3.6. This has &O ben shown for the other type of source/sink configuration in the domain. For example, Figures 3.7 and 3.8 illustrate the ciifference for a control-volume pressure-source/sink con- siderations. The Iack of mass conserving equation WUresult non-physical solution in domain.

AU the resdts of this section (up to here) were obtained by employing the convecting momentum expression of Method 1 into the continuity equation and considering a=l, Eq.(3.56). The depicted figures show that mass is constant in passing through pressure-source/sink while pressure is not constant passing through a mass-source/sink. The use of momentum convecting equations in continuity equa- CHAPTER 3. ONEDIMENSIONAL INVESTIGATION AND RESULTS 68

Pressure Disiribution I . T I

'8 6.1 0.2 0.3 014 8.5 0.6 07 0.8 0.9 ! Node and i.p distnbubon

Figure 3.4: The &ct of control-volume mass-source/sink on pressure and velocit y distributions.

equation removes the checkerboard problem and res trict s oscillations to t kee nodes or less.

There are also many otha results with employing Method II and Method III of Section 3.5.3. They show satisfactorily results in removing the pressure checkerboard problem. These results have been compared with each other. Genedly, Method 1 shows better or similar results compared with Method II and Method III. For example, Figure 3.9 shows the pressure distribution cornparison when control- volume pressure-sourcefsink is located in the domain. The Method II with a=2, Eq.(3.65), shows overshoot and undershoot in the neighbouring nodal &es but Method 1 wit h a=l, Eq.(3.56), shows just one stronger overshoot downstream of the source or si&. Three nodal points are affected by the sink/source in the Method XI while this is decressed to two nodes in Method 1. The reason for the difference in 18 0.1 02 63 0.4 os 0.6 -0.7 0.8 o.~ 1 Node and i.p dibubon

Figure 3.5: The dect of element mass-source/sink on pressure and velocity distributions without using convecting momentum equation.

Mass Distribution 2.5 .

1 0.1 02 0.3 0.4 0.5 0.6 O.? 0.8 0.9 t

'% 0.1 6.2 6.3 0.4 6.5 0.6 0.7 0.8 0.9 ! Node and i.p dîbutkn

Figure 3.6: The effect of element mass-source/sink on pressure and velocity distributions. CHAPTER 3. 0NEDIMENSIONA.L INVESTIGATION AND RBSULTS 70

Mass Disîrittutkn 2' 2' forrodeO -for i.p x

hPressure Datributh

Figure 3.7: The effect of control-volume pressure-source/sink pressure and velocity distributions without using conuecting momentum equation.

'4 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Node and i.p dislnbum 1

Figure 3.8: The &ect of control-volume pressure-soarce/sink on pressure and velocity distributions. CHAPTER 3. OMUXMENSIONAL INVESTIGATION AND RESULTS 71

Figure 3.9: Comparing the numerical results of Method 1 and II using control- volume pressure-source/sinlr in domain. solutions returns to the connection of momentum components at integration point to its neighbouring nodes. In an upwind scheme, the integration point is dected by upstream values while in central differencing scheme, the effect is fiom both ups tream and downs tream values.

3.7.2 Compressible Flow

Aere, the shock tube problem has been selected to present the resdts of the one- dimensional formulation for compressible flow. The selection of the shock tube problem as a test case is due to the availability of its analytical solution which enables us to examine the accuracy of the numerical solution. On the other hand, this transient problem includes both transient fiow featntes and a wide range of Mach nnmber, i.e., a moving normal shock wave, expansion waves, a contact discontinuity, CHAPTER 3. ONEDIMENSIONAL INVESTIGATION AND RESULTS 72 and having subsonic, transonic, and supersonic regimes.

The shock tube geometry and wave pattern is shown in Figure 3.10. The pressures at the left and right of the diaphragm are taken as lOOOkPa and 100kPa, respectively. The temperature is uniform tkoughout the shodr tube at 25 OC before nipturing the diaphragm. Gas properties are e. = 720 J/kgK, R = 287.0 J/kgK, and 7 = 1.4. To the left of the contact line, gas expansion causes a reduction in

Contact Surface

Figure 3.10: Shock tube problem and its wave pattern. temperature, whereas to the right of the discontinuity, the compressed wave raises the temperature of the gas. At the contact line, there is a discontinuity in the temperature profile and hence the density profle.

The resdts show the mass flux, pressure, temperature, and density distributions throughout the shock tube 500 ps der nipturing the diaphragm. The exact solution is superposed in ail figures. 201 nodes are chosen with a the step of 0.7 pu. The pressure and temperature are nondimensionalized by lower pressure aide CHAPTER 3. 0NEDIlMENSIONA.L INVESTIGATION AND RESULTS 73 values and the initial temperature of the shock tube, respectively. The convergence criterion in each time step is checked for all nodes,

where i indicates the node number and e=lOO'.

The shock tube problem is examined both for different Courant numbers and for Methods 1, II, and III. Figure 3.11 shows the results employing the mass conserving procedure of Method 1. The moving shock wave is captured within a few nodes, but

0.5 Shodc Tube Axis (m) Shed< Tube kjs(m)

Figure 3.11: Shock tube results for Method 1 with a =1, -0.4, and 201 nodes. some overshoot is seen downstream of the shock wave. The Courant number for this case is 0.4. In order to find the limits of stability for our implicit algorithm, the time step is gradoally increased to increase the Courant number. Figures 3.12 and 3.13 show the results with larger time step sizes, i.e., At=l2ps and At=17ps, respectively. They show that the method is stable for higher Courant numbers of CHAPTER 3. ONEDIMENSIONAL INVESTIGATION AND RESULTS 74

4- 4- as 1 Shock Tuba Axii (m)

Figure 3.12: Shock tube results for Method 1 with a=l, 60.7, and 201 nodes.

OB*- OB*- 0.5 1 Shodc Tub, ki (m)

Figure 3.13: Shodt tube results for Method 1 with a=l, 61.0, and 201 nodes. CHAPTER 3. ONEDIMENSIONAL I'STIGATIONAND RESULTS 75

C=O .7 and G1.O, respectively. The inaease in Courant nnmber does cause more smearing around discontinaities because the transient details are not captured accu- rately in fnlly implicit methods with large time steps. No bits were encountered for higher Courant numbers. There are results available for Courant number of e6.3 which are not presented hem.

In the second stage, compressible flow is tested for two more test cases involving Method II and Method III. The results have been depicted in Figures 3.14 and 3.15 for Method II with a=l and for Method III with a=2, respectively. The results of Method II are identical with the results of Method 1 because their formulation become identical for the defined a's in this test case. It is interesting to note that discontinuities are predicted within a fewer number of nodes in Method III, but at the cost of higher undershoot and overshoot around discontinuities.

The final stage is to study the effect of mesh refinement. Following the results presented in Figure 3.11, G0.4 is selected as constant and At is changed for a total number of 151 nodes to maintain Uk0.4. The resdts are seen in Figure 3.16. As is expected, sharp changes of the parameters are smeared and there would be less accuracy than the case of Figure 3.1 1 with 50 more nodes.

As the results show, all methods remove the cheekerboard problem and provide the desired coupling between pressure and the momentum components. These preliminary result s show good agreement wit h the t heoretical solution although neit her ar tificial viscosit y nor ot her overshoot treat ment is explicitly considered in the formulation. Moreover, no smoothing or damping function has been used to reduce oscillations. The results of this section confirm the success of the one- dimensional formulation for treating high speed flows. CHAFTER 3. ONED~NSIOnrALSTlGATlON AND RESWS 76

mu* 15001 1

t a5 1 Shock Tube Axis (m)

Figure 3.14: Shock tube, using Method II with a=l, -0.4, and 201 nodes.

Figure 3.15: Shock tube, using Method III with a=2, G0.4, and 201 nodes. CHAPTER 3. ONEDLMENSIONAL INVESTIGATION AND RESULTS 77

0.5 1 Shodc Tubi AxI (m)

Figure 3.16: Shock tube results for Method 1 with a=l, e0.4, and 151 nodes.

3.7.3 Cornparison Between Velocity and Momentum For- mulations

As the final step of the one-dimensional investigation, the performance of the me mentum component formulation is compared with that of the velocity component formulation. In this way we are able to investigate some of the potential adïan- tages of the momentum component formulation. One advantage relates to the benefits of using mass flux variable instead of the velocity-variable, Section 2.4.2, for flows involving discontinuities, and mot her relates to the benefit s of simplifying the difficult treatment of nonlinearities, Section 2.4.3. The shock tube problem and the quasi-one-dimensional converging-diverging nozzle problem are two one- dimensional test problems which can be used to investigate the potential advantages of the momentun-variable formulation Li high speed flows with shocks. In CHAPTBR 3. ONEDIMENSIONAL INVESTIGATION AND RESULTS 78 this section, we pursue this investigation for the shock tube problem which is a more difficult problem as was mentioned in Section 3.7.2. The initial conditions, the gas propesties, and the nodes distributions are defined as betore. This investigation is restricted to isothermal 0ow in order to provide a clearer cornparison between the two formulations. Equation 3.80 is used as a measure of convergence for the pressure.

Figures 3.17 to 3.20 present the results obtained by both the vdocity-based and the momentum-based formulations. They show the distributions throughout the shock tube 500p after rupturing the diaphragm. The convergence criterion had been set c=lO-a for both formulations in this test. The velocity-based results have been obtained by modeling the equations as presented in Section 3.6 which represents the work of Karimian and Schneider [65] except for the exclusion of the damping mechanisms of th& formulation. They have presented a detailed study of their approach in solving the shock tube problem. The momentnm-based formulation has been treated in a manner which provides more consistency with the velocity-based formulation. For example, the momentum convection term in t his formulation, Eq.(3.12), is linearized very similar to the one in the velocity-based formulation, i.e. kt=: and k"=O.

Several conclusions can be derived fiom this study and fiom examination of the presented figures. Generally speaking, the velocity-based fosmulation sders from severe oscillations in the vicinity of the shock. Experience showed that there was also a maximum Courant number for the velocity-variable formulation that enabled solutions to be obtained. This value, =O. 1, is the one for which the presented results have been obtained. Above this value, the method was unstable and diverged. Howeva, the momentum-variable method converged for higha Courant numbers, up to G2.2, although the accuracy of the solution was degraded with inaeasing CHAPTER 3. ONEDIMENSIOEIAL INVESTIGATION AND RESUCTS

mass flux distribution I v .

Figure 3.17: The comparison of mass flu distributions between velocity-variable and momentum-variable formulations, c= IO-?

velocity distribution SOOl 1

Figure 3.18: The comparison of velocity distributions between velocity-variable and momentum-variable formulations, s-10-? pressure diiributbn 10

0 - momiMiW

8 - nbdlrw

7 -

E 6-

4 -

3 -

2 -

1 O a1 02 0.4 as as O.? 0.a a0 1 shock tube lengîh(m)

Figure 3.19: The cornpaison of pressure distributions between velouty-variable and moment um-variable formulations, a= IO-'.

Figure 3.20: The cornparison of density distributions between velocity-variable and momentum-variable formulations, c= IO-'. CHAPTER 3. ONE-DLMENSI0NA.L DM3STIGATION AND RESULTS 81 the Contant number. The presented result s for the momentnm-based formulation were obtained using e0.35.

The distributions of the fiow parameters in this isothermal study show fewer discontinuities than the corresponding distributions in Figures 3.11 to 3.16. This has been shown in Figure 3.21 by pdorming a rough comparison between the isothermal and non-isothermal exact solutions. In this figure, we are concerned ody with the general distribution of the parameters rather than th& exact values and positions. As seen, a constant temperature field resdts in no discontinuity

Mess flux

Figure 3.21: A rough comparison between isothermal and non-isothamal exact solutions in the shock tube problem. in the form of a contact surface. However, there is still a sharp discontinuity with the moving shock which appears in ail distributions induding the mass flux distribution. These discontinuities are due to the nature of the problem which is a highly transient one. As a reminder, we were concerned with the two potential CHAPTER 3. 0NEDLIMENSIONA.L LNWSTIGATION AND RESULTS 82 advantages of the momentum-variable formulation, i.e. the constant mass flux through àiscontinuity and the nonlinearity treatment. Figure 3.21 shows that the number of discontinuities in the mass flux distribution is never less than the number of them in the velocity distribution. In other words, whereas F is indeed constant across a shock in steady flow, here it is not. This problem therefore demonstrates that the 0th- aspects of the momentum-based formulation, related to hearization reqnirements, have resulted in superior results to those from the velocity-based formulation.

One more step was taken in this comparative study in order to cheJ the stability of the two formulations for meeting lower convergence aiterion. In this regard, the convergence criterion for the pressure was decreased from IO-' to IO-'. In this case, the vdocity-based formulation diverged for the previous Courant number, G0.1. The maximum Courant number which enabled solutions to be obtained with this lower convergence criterion was G0.03. The results for this Courant number have been illustrated in Figures 3.22 and 3.23. They show much stronger oscillations in the domain comparing with the previous results. The results for the momentum- based formulation have been obtained for Courant number 0.35, as before, and an extremely low convergence criterion of IO-'. The decrease of the convergence criterion fiom IO-' to increased the average number of iterations per time step from 5 to 14.

A onedimensional investigation was pdormed in this chapter. The main purpose of this investigation was to ensure that the developed momentum-component pr* cedure works well in its one-dimensional form and that all proposed objectives have CHAPTER 3. ONE-DIlMENSIONAL mSTIGATION AND RESULTS 83

pressure distribution

Figure 3.22: Comparison of pressure distributions between velocity-variable and momentun-variable formulations, a= IO-'.

velocity distribution . 1

-'O% 01 02 03 0; 0; 0; O;, 9. 0; ! shock tube length(m)

Figure 3.23: Cornparison of velocity distributions between velocity-variable and moment--variable formulations, a= IOœ5. CHAPTER 3. OWDLMENSIONAt IiVVESTIGATION AND RESULTS 84 been satisfact orily achieved. In t his regard, the conhol-volume-based formnlat ion was appiied to the one-dimensional governing equations. The related integration point operators wer- derived based on incorporating the correct physical idoence of the flow and other relevant schemes. The simplified forms of the derived expressions were tested for special flow cases with dSerent Peclet and Courant numbers for which the results illustrated a behaviour consistent with the physics of the flow. The pressure checkerboard problem was removed using convecting integration point operators for momentum components. In this regard, several integration point expressions were derived for use in the continuity equation which showed satisfactory results in coupling the velocity and pressure fields. The ability of the method to remove the pressure checkboard problem was tested by putting a mass/pressure source/sinL in one-dimensional compressible flows. The method worked effectively. In compressible flow, the neu formulation was tested for the shodc tube problem which has many features of high speed compressible flows induding moving normal shock and expansion waves. There was no CFL number limit observed for the present ed implicit algorithm.

A direct cornparison of the velocity-based and momentum-based formulations was performed. It was shown that the velocity-based formulation produces severe spatial oscillations if an explicit damping mechanism is not employed. These osdations subsequently resulted in increasing the number of iterations per tirne step and lowering the accuracy of the solution. Then, it was condudecl that the momentum-based procedure produced more accurate and stable solution than the velocity-based procedure without damping.

The results of this chapter support the objectives of the current research and enable us to move towards extending the one-dimensional formulation to a ho- dimensional momentum-component formulation. Chapter 4

Computat ional Modeling in Two Dimensions

4.1 Introduction

The preliminary investigations and results for one-hensional flow were accomplished in Chapter 3 and shown to be satisfactory. Here we extend the proposed approach to two-dimensions. In this regard, the necessary steps of the discretba- tion procedure for the two-dimensional Navier-Stokes equations are presented in this chapter. These steps are similar to those taken in Chapter 3.

After the introduction, Section 4.2 provides definitions and descriptions which are used throughout this chapter. In order to discretize the tw~dimensionalgov- erning equations, a control-volume-based finite-element approach is employed to integrate them over the control volumes in Section 4.3. The discretized equations which axe derived in this manner require the evalaation of the dependent variables at control volume sudaces. Thus, the necessary integration point operators ate CHAPTER 4. COMPUTATIONAL MODELING LN TWO DIMENSIONS 86 derived in Section 4.4. This is where some important issues of convection-di&ision modeling and velocity-pressure coupling are discussed. The convecting momentnm equations are obtained in Section 4.5. The resdts of previoosly-mentioned three sections are assembled in Section 4.6 where the element stifiess matnx is built. Finally, the techniques used to invoke boundary conditions are explained in Section 4.7.

4.2 Preliminary Definitions and Descriptions

In the following sections, the details of the discretization technique will be presented. In order to unify all definitions and conventions which are used during discretization, it is helpfd to present them in an introductory section. This is accomplished in the current section.

The discretization is started by integrating the governing equations over control volumes. In the employed control-volume-based approach, the procedure of integration is accomplished in an element-by-element mamer. The process, within all elements, is started fiom SCVl (Su b Control Volume 1) of the element and extends to SCV4 of that element, Figure 2.1 in Section 2.2. In order to distinguish the sub-control-volumes, we use the index of i, i = 1.. .4, at the end of SCV to identify the SCV in question, Table 4.1. Furthamore, i is the relevant node namba in that SCV. There are four sides in each SCV only two of which are coincident with the correspondhg control volume edges. For example, SS4 and SSI are the edges when SCVl is treated. For an arbitrary SCVi, these sub-surfaces are Sli and S2i for which the numbers represent the sub-surfaces when a counter-clockwise rotation is selected around the center of the element for traveling fiom one sub-sudace of the SCVi to another. Table 4.1 also shows the arrangement of Sii and S2i for CHAPTER 4. COMPUTATIONAL MODELIlVG IN WODIMENSIONS 87

- SCVi SCVl SCV2 SCV3 SCV4 l Sii,SSi SS4,SSl SS1,SSZ SS2,SS3 SS3,SS4

, $Si Jsss, + /sSn SS,, + Is,, is,, + rssm Jssm + jsss, Yi Vscvl Vscvt YSCV~ vscvr * - Table 4.1: Abbreviations and parameter definitions. different SCVs of an element. The integration over these two sub-surfaces is briefly addressed by a single digit number index. For SCVl, it is given by

Similar defini tions have been tabulated for other SCVs in Table 4.1. Other parameters of the SCVi may be identiiied directly by adding the subscripts i to them. For example, the volume per mit depth of SCVi is shon by

Approximation of the governing equations will algebraicaily require the modeling of each of the operators in terms of at most four nodes and four integration points in each element . This results in tao sets of 4 x 1 array of unknowns which d repeatedly be encountered in this chapter. The iirst set represents the magnitude of dependent variable or anknowns at nodes, i.e., {P), {F), {G}, and {T), and the 0th- set represents them at integration points, i.e., @}, {f), {g), and {t). The latter are not nodal unknowns and must therefore be related to these nodal dependent variables.

Generally speaking, there are four major govaning eqaations, four sub-conkol- volumes, four nodes, four integration points, etc. which make their tracking corn- CHAPTER 4. COMPUTATIONAL MODELIEIG IN TWO DIMENSIONS 88 plicated. In orda to have an organized procedute, many us& definitions are presented here. For example, the results of integration over each SCV are required to be stored in arrays. Special combination of subscripts and superscripts are used to recognize the identity of the elements of the arrays. They are generally written as

The upper and lower cases for [A] and [a] mean the matrices multiply by the array of nodes or of integration points of an element ,respectively. The notation spl refers to the related govaning equation, i.e., p:continuity, fiz-momentum, g: y-momentnm, and t:energy. The notation sp2 stands for the multiplier array to the matrix, i.e., p, f, g, and t for P, F, G, and T arrays, respectively. The notation sp3 means to which term the matrix belongs, e.g. 0: transient term, c:convection term, d:dinusion term, p:pressure term, and etc. The notation sbl stands for SCV number in question within element, and sb2 counts either the element node numbers, if the coefficient is upper case, or integration point number, if it is lower case.

A well-posed discretization will require the representation of integration point values in terms of nodal ones. This procedure wiIl result in matrices similar to Eq.(4.3) which are identified by [Cl and [cl, i.e.,

All subscripts and superscripts are identically defined as those in Eq.(4.3) except for sbl which refers to integration point number now. The array of known dues in right-hand-side of elemental matrices, Eq.(4.3), and integration point matrices, Eq.(4.4), are identified by {A) and {C), respectively, i.e., CHAPTER 4. COMPUTATIONAL MODELUVG LN TWO DIMENSIONS 89

Thei. subscripts and superscripts have the same definition as before in tams of spl, sp2, sp3, sbl, and sb2.

4.3 Discretizat ion of the Governing Equations

The generai form of the governing equations was introduced in Section 2.1 by Eq.(2.18). Considering no source terms, these equations can be integrated over SCVi of an arbitrary &ment

Using the divergence t heorem, the volume int egral of space derivatives is replaced by a surface integral which is evaluated over the surface of SCVi

where d3 is the outward normal vector to the surface, Figure 2.3. Integration on surfaces is broken into two sub-surface integration accordtig to Table 4.1

/ (IP;.+T~)*~s+L~(R~+T?)-~~ Sli where Sli and S2i indicate the integration point number on the sub-surfaces. Since the method is fdy implicit, d terms except the transient term are evaluated at the advanced time and the transient term is approximated by a lumped mus approach, i.e., $ at each SCV is approximated by the nodal value of its correspondhg node CHAPTER 4. COMPUTATIONAL MODELING IN TWO DIMENSIONS 90 where Ji is the Jacobian of transformation and represents the volume of the subcon- ho1 volume per unit width, Eq.(A.15). A lineat variation of the integral argument is considered over eaeh sub-surface integrals. As a result, the value of the surf'ace integral is approximated by the value at the mid-point of that sub-surface, i-e.,integration point, times the area of the sub-surface. By this mid-point approximation, the arguments of the integrals are taken out of the integral and the areas of the sub-sudaces are computed using Eqs. (XE), i.e.,

Considering this mid-point approximation and substitution of Eq.(4.9) into Eq.(4.8) results in the final conservative discretization form of the governing equations for scvi

where the subsaipt i of iY refers to SCVI. This general equation consists of aU transient, convection, and diffusion flux terms of the four governing equations.

Recdling the definition of 7,Ç, 'R, and 7,Eqs.(2.20 and 2.21), shows that there are two major obstacles to solve Eq.(4.11). The first obstacle returns to the nonlinear nature of the system of equations. In order to use linear algebraic equation solvers, these equations must be iinearized. The second obstacle is the location of the unknown fluxes and flows. As seen, the locations of the unknowns are integration points which are not out nodal locations and they must somehow be related to the main nodal values. These difficulties are treated in the present and the following sections. CHAPTER 4. COMPUTATIONAL MODELING IN TWO D~NSfONS 91

4.3.1 Conservation of Mass Equation

The discretized form of conservation of mass could be extracted fiom the general form of Eq.(4.11) and re&g -the definition of F and Ç nom Eq.(2.20)

The transient term is a non-linear term because e is not a major dependent variable and needs to be lineariized properly. The density in incompressible flow is constant, Eq.(2.16), and does not need any treatment. The equation of state is used to convert density in compressible flow. Appendix B presents methods for densi ty linearization. Regardhg the results of t his appendix, Eq. (B.5) is selected to linearize e with respect to bath P and T dependent variables

Thus, the transient term is written as

As seen, the results are easily converted to the compact sununation form. It should be noted that three of the fou components of each summation are zero in this general form, i.e., CHAPTER 4. COMPUTATIONAL MODELING IN TWO DIMENSlONSIONS 92

Now, the transient term, Eq.(4.14), is written in snmmation form for an arbitrary sub-control-volume of i as

A similar procedure could be repeated for the mass flux terms of the continuity equation. &cd Eq.(4.8) and consider the bracketed terms of Eq.(4.12), the summation form becomes

Here, two of the four elements in each siimmation are zero, i.e.,

(AS.),, j = Sli A j=s2i O eise

(ASu),, j = Sli j = S2i O else

The two discretized equations, Eq.(4.18) and Eq.(4.19), can now be assembled and written in the usual matrix form for an arbitrary sub-control-volume of i

It should be noted that the above equation is written only for one portion of a four-control-volume element. It is not expected that it conserves the mass within CHAPTER 4. COMPUTATIONAL MODELilVG IN TWO DLMENSIONS 93 that sub-control-volume of i. The conservation of mass is only valid for a whole control volume which consists of an assemblage of four suh-control-volume equcl- tions, Figure 2.2. If all four sub-control-volume equations of an element are put togetha they take the foIIowing matrix form:

where the brackets are 4 x 4 square matrices and the braces are 4 x 1 colnmn vectors. The rows of arrays represent the sub-control-volume numba and the columns represent the node number.

Notice that Eq.(4.23) has discretized the mass flux terms into an algebraic expression involving the integration point variables as wdas nodal Mnables. This expression is not complete at this stage and needs to be modified. This modification is done in Section 4.5.

4.3.2 Conservation of Moment um Equat ion

Discretkation of the momentum conservation equations is much more complex than that for the continuity equation. The complexity of the equation &ses due to the appearance of difïerent types of nonlinear terms, and from the large number of them. The one-dimensional discretization of Section 3.3.2 will facilitate the procedure of discretization in this section. The transient term does not need any linearization and it is easily discretized by plugging the definition of $ fiom Eq.(2.19) into Eq. (4. Il), usiog a mas-lumped approach, CHAPTER 4. C0MPUTATlON.L MODELING IN TWO DIMENSIONS 94

A s-ation form similar to Eq.(4.18) could be arranged for an arbitrary sub- control-volume i

where

As the nat step, the convection terms are treated. In this regard, the appr* priate terms of 3 and Ç, Eq.(2.20), are substituted in the integral form of the convection terms

We fkst treat the pressure integration term which is linear. Considering the previous procedure, we write

Using our previous definitions, this equation is reformed in general sammation form for an arbitrary sub-control-volume i as

where the coefficients of a{? and those of 4'imin Eq.(4.20) are the same, i.e., CHAPTER 4. COMPUTATIONAL MODELLNG IN TWO DlMENSIONS 95

As seen, the pressure term is &O represented by integration point variables which in tum shodd be represented in terms of nodal variables. This WUbe accomplished in Section 4.4.1. The first integration on the right-hand-side of Eq.(4.28) can be expanded for

As seen, the discretized equation is nonlinear and needs to be linearized properly. Appendix C has a comprehensive stndy on different methods of linearization for the convection terms of the conservative momentum equation. Providhg a simpler approach at this stage, Eqs. (C.10 and C.ll) with k' = O are nsed to linearize the nonlinearit ies , i.e.,

where ü and ü are calculated explicitly fiom known values of the previous iteration which will be explained later in Section 4.5. Using these linearizations in Eq.(4.32), we obtain 2 (pu;+ puj) dS [[uf (AS,) + üf (AS,)], (4-34) k=1 The general summation form for an arbi trary sub-control-volume i is

where the coefficients are CHAPTER 4. COMPUTATIONAL MODELING IN TWO DIMENSIONS 96

As the second step, integration point values which appear in lineaxized terms have to be calculated as a funetion of the nodal variables in the element. The appropriate equations for f and g are derived in Section 4.4.1.

To complete the momentum conservation equation, the dihion terms must be treated. Modeling of the diffusion terms is relatively more routine than the convection terms due to their elliptic nature. Following the previous procedure, the right-hand-side of Eq. (4.11) could be extended to the momentum equations using the definition of 'R and 7 fiom Eq.(2.21). The result is

Looking back to the definition of stress terms in Eqs.(2.6 and 2.7) shows that they are nonlinear in terms of momentum component variables. In the one-dimensional investigation, Section 3.3.2, the linearization scheme of Appendk D was selected to linearize the 2 term. In the two-dimensional difhsion terms, the essence of nonlinearity is the same although they are more plentifid. As before, we use the approach presented by Eq.(D.2) in Appendix D to linearize the velocity differential forms. Employing t his linearization scheme to the stress terms of Eqs. (2.6 and 2.7) results in

(4.38)

(4.39)

(4.40)

The second terms in ail braces vanish in the incompressible limit. These terms are now lagged and calculated explicitly from the knoum dues of the previous CHAPTER 4. COMPUTATIONAL MODELING LN TWO DIMENSIONS 97

iterations. This results in a source term on the right-hand-side of Eq.(4.8) in mm- pressible flows. To reveal the elIiptic nature of difbion, dl active and inactive differential terms are modeled thrmgh differentiation of the finitdement shape hctions for the hear quadrilateral elements, Eq.(A.3). The fonowing results are obtained after rearranging and sorthg active and inactive terms together:

Now, Eqs.(4.41 and 4.42) are substituted in Eq.(4.37). The resulting equation can be rearranged and written in general summation form for an arbitrary SCVi

where the coefficient of ~{f"and are

2 4/i B Nj A{!' = - (AS.) + --(AS*)]P air Ski

2 A{!$ = - 2p (AS.) + -%(AS,,)] p 62 SL~ CHAPTER 4. COMPUTATIONAL MODELING IN TWO DXMENSIONS 98

Non, we can assemble all the terms of the momentum equation. In this regard, Eqs.(4.25, 4.30, 4.35, and 4.44) are plugged into E44.7) and the result is

which could be cast into the matrix form as

[A"' + Af fd]{~) + [Afgdj{~)+ [affq{f) + [a*] Cp) = {AI' + (4.49)

Similar to previous sections, t hese equations are not well-posed because they involve integration point values. Closute for these terms wiU be fmed in the next sections. AU discussions in this section relate to the discretization of the 2- momentuxn equation. A similar procedure is applied for the y-momentum equation. The results of this would be an equation similar to Eq.(4.49). Comparing with this equation, it is written

where the elements of the matrices could be estimated by cornparhg the equident terms of two momentum equations and the resulting elements of the z-momentum equation mat rices.

4.3.3 Conservation of Energy Equation

The discretization of the energy equation is not as straight forward as it was for mass and momentum equations. There are a variety of reasons behind this which cause the study of the energy equation to be not so routine as for the others. Among these reasons, one is the larger number of terms and the other is their higher complexity. This complexity however causes more complex nonlinearit ies for t hose CHAPTER 4. COMPUTATIONAL MODELING IN TWO DLMENSIONS 99 terms. These complex nonlinearities can be treated using difkrent techniques of linearization. Howevr, more study and experience is needed to Myassess the pros and cons of different possible linearization. A poor linearization may cause oscillation in the solution which in turn may lead to divergence or slow convergence. Ail these reasom cause the study of the energy equation to be more diffidt and ambiguous. Contrary to the important terms, there are a number of nonlinearities which are not so prominent. This permits us to ignore the active role of many terms and dculate t hem approximately using the known values from previous iterations.

We start the discretization by treating the transient part of the equation. Con- sidering Eq.(4.9) and the defmition of Ji from Eq.(2.19), the transient part of the equation is mitten as

(4.51)

There are difkrent possible met hods t O derive the linearized form of this equation. Appendix F presents a few of them. Regmding the comments of the Appendix and the one-dimensional procedure, Section 3.3.3, we use Eq. (F.2)to linearize the transient term

This results in a general summation form for an CHAPTER 4. COMPUTATIONAL MODELING LN TW;O DIMENSIONS 100

In the second step of disaetization, the energy convection tamsare discretized. Combining the proper components of 3 and Ç from Eq.(2.20) and the convection part of Eq.(4.7), we obtain

a 2

SimiIar to the momentum convective terms, this term is nonlinear and must be linearized in terms of momentum component variables. Appendix G presents different schemes of linearization for the above nonlinearities. At this stage, they are linearized using the scheme presented by Eq.(G.3). The employment of this scheme y ields

ü - (fif + üg + 2cppt) (AS,)] 2 ski This equation is reformed to a general snmmation form for an arbitrary SCVi CHAPTER 4. COMPUTATIONAL MODELLNG IN TWO DIMENSIONS 101 where the co&cients of the matrices are obtained fiom j = Sli j = S2i else

j = Sli j = S2i else

j = Sli j = S2i (4.63) else Despite a full discretization, the unknowns of this equation still need to be related to the main nodal variables. This is accomplished in the next sections. The last part of the discretization process is the modeling of the energy viscous work terms. As mentioned earlier, the elliptic nature of diffusion facilitates its treatment. To treat the diffusive conduction terms, Eq.(4.7) is recalled and the conduction terms of R and 7 in Eq. (2.21) is plugged in as

Using the definition of q, and q, fiom Eq.(2.9) yields BT aT (n.g+quj)md?? = -LE[z(~~.) + -(AS.)] (4.65) k=l s~ Ski The derivatives of the finite-element shape functions are employed to describe the diffaential forxns, i.e., 4 2 SNj BNj (q= ;+ ,f ) 8 a -k [-a;(~~=)+ Sy(~~u)] Tj (4.66) j=l k=l Ski CHAPTER 4. COMPUTATIONAL MODELNG DU TWû DIMENSIONS 102

The general form of this for an arbitrary SCVi is

where 2 &!k = k=l Skà For the remainder of the diffusion terms, we use again the definition of R and 7 hmEq.(2.21) and plug its energy part into Eq44.7). The result is

The nonlinearity of the viscous work terms is much complicated than the diffusion terms in rnomentum equation. This is due to the velocity components which rnultiply the stress components. At the fist stage of the linearization, ali these veiocit y components are evaluated using lagged values fkom the previous iteration and the treatmerit of the remainder of the nonlinearities are sida to those for the moment- equations. We start discretizing by plugging the linearized stress tas, Eqs.(4.41-4.43), into Eq.(4.69). The resulting equation can be rearranged for the active and inactive variables. The general form of the resulting equation is presented in summation form for an arbitrary SCVi as

where the matrix components are dehed as CHAPTER 4. COMPUTATIONAL MODELZNG 12V TWO DIMENSIONS 103

ü2aNj üü 8Nj 2U5 aNj 4ü2 8Nj] (ASy)) - -- +----- ëj (4.74) p 82 3p 82 3p By ski

Now, ne can assemble al1 the terms of the energy equation in Eq.(4.7). In this regard, Eqs.(4.53, 4.60, 4.67, and 4.70) are plugged in Eq.(4.7) and the resdt is

which can be cast into the matrix form as

The treatment of integration point values is necessary as it was in preceding sub- sections. The next section WU provide this treatment.

4.4 Integration Point Equations

The idea of connecting the integration point values to the correspondhg nodal values has been largely investigated in control volume methods. In Section 4.3, we diacretized the governing equations in a control volume manna. The resulting CHAPTER 4. COMPUTATIONAL MODELING IN TWO DlMENSIONS 104

equations indudecl the may of integration point unknowns which are not the major unknown nodal variables in our formulation. In order to have the discretization complete, it is required to represent the momentum components, pressure, and temperature at the integration points of an element in terms of their neighboring nodal values within that element. In Section 4.3.2, the nonlinear forms of 7Z and 7,Eq.(2.21), were directly treated by invoking the elliptic nature of diffusion. On the 0th- hand, time dependent terms were directly computed from nodal dues. However, convective tenns are left to be evaluated in a special manner because th& treatment is not as direct as it was for the diffusion terms. Roughly, about (80 to 90)% of the difficulty in modeling of momentum equation terms returns to the complexity in modeling of convection terms, Patah [66]. Central-diffaence schemes in finite-difference approaches and bilinear interpolation in hite-element approaches are the easiest ways to treat the convectivz iniegration point values wit hin an element, however, t hey are limited to diffusion-dorninated flows where Peclet number is less than 2.

For convective dominated flows, it is better to use upwinding-based methods which consider the higher influence of the upstream nodal values at integration points. Figure 4.1 schematicaily shows how the upstream influence should be re- garded in evalaating convective integration point quantities. A pure upwind scheme works for all Peclet numbers, however, the accuracy of the solution is diminished. Raithby [67] has provided a comparative study on treating the convection terms by a number of upwtid schemes and shows that they produce fbdiff'usion. Raithby

[68] has &O proposed a skewed upwind scheme which considers the real direction of the flow and mnsequently reduces the false-diffusion. The basic attemp t in most recent schemes is to mode1 the integration point values, &,, by upstream values, CHAPTER 4. COMPUTATIONAL MODELIN IN TWO DMENSIONS 105

Figure 4.1: A schematic influence of npwind point. dUp,and then correct it by A&,

The essence is that the connection is not only based on complex mathematical fnnctions but also on the physical interpretation of the governing equations. Much dort has been expanded to determine be t t er and simpler interpretations induding al1 necessary influences. There are three basic points which must be remembered in determinhg integration point operators; 1-efFect of directionality of the flow; %source (including pressure) term efFects; and tthe dinusion efFects. The basic idea in modeling convective quantities is to treat the integration point dues by the correct influence of upstream values. This is why different schemes like hybrid, power-law, upwind, exponential, etc. have been established over the years for their treatment . The folloving sub-section present the methods of deriving the required integration point equations for different variables. CHAPTER 4. COMPUTATlONAL MODELING IN TWO DIMENSIONS 106

4.4.1 Momentum Components

In this section, we derive f and g integration point equations in an appropriate maMer which considers the physics of the problem. More recently, control-volume- based methods have been concemed with the physics of the problem rather than complex mathemat ical approximations. Following this, Schneider and Raw [IO] proposed a new scheme, Correct-Physical-Influence Scheme, which includes all the physics and relevant couplings for each variable. They have derived an algebraic approximation to the differentiai equations at each integration point. In Section 3.4.1, it was shown that this new scheme yields a physicdy well-behaved solution in the one-dimensional study. Following the one-dimensional investigation, the tw* dimensional moment um integration point equations are derived by treating the non- conservative form of the momentum equations. In this way, the convection terms in the conservative form of the momentum equations are broken into two terms, combined with continuity equation and hally written in a streamwise direction, Appendix H. The following momentum equations are the result of this procedure: Of f 8~ 8~ - +Vw-a -,.pu= -- + u(- + V&-)a~ + Viscous Terms (4.78) de 8s dz ae 0s

ap ap -as Vu- as - p~2v= -- v(- V--)a~ Viscous Terms (4.79) ae + as ay + ae + as + where Vw = I/w.Viscous Te- stand for the viscous terms which are not induded at this point. In this section, fip is derived fiom Eq.(4.78) and the results are similarly extended to which must be equally derived from Eq.(4.79).

First of all, the transient term in Eq.(4.78) is written in backward form respect to time CHAPTER 4. COMPUTATIONAL MODELING IN TWO DIMENSIONS 107 where the subsaipt i denotes the integration point number. This disaetization can be ast into matrix form for an arbitrary integration point of i

Here the notation of ipi is switched to i which indicates integration point. The coefficients in question are

The key in this method is found in the convection term which has been written in the streamwise direction. This form provides the correct direction of upwinding in the streamwise direction. The one-dimensional investigation of this equation showed the flexibility of this formuktion to predict the correct f for highly dinu- sive and convective flows, Section 3.4.1. The convection term is upwinded in the streamwise direction as

where Lw and f, are iliustrated in Figure 4.1 for ipl. The upstream location, up, is found by interseeting the extension of the streamline direction at integration point with the edges of the same dement. The value for f, is interpolated between the two adjacent nodes which are nodes 2 and 3 for ipl in Figure 4.1, i.e., CHAPTER 4. COMPUTATIONAL MODELING IN TWO DIMENSIONS 108 where (F,), and (Fd)+ refer to the nodal values of F at right and lefk of the upstream point 'upn, respectively, when "up" point is watched from integration point i. Using this dennition, a general form for integration point i is given by

where the matrix coefficients are

Here, the magnitude of c&" depends only on the flow direction which is obtallied by using Eq. (4.85).

The Mass- Weight ed-Skew scheme of Schneider and Raw 1691 is another accurate method for treating the convection terms. It reduces the possibility of negative coefficients in the formulation. This scheme models the local directionality of the flow in the rnanner that ensures positive coefficients in the convection terms of the discretized control volume equations. It finds the flow direction within the element and cornputes Lw and f, based on the values and directions of flow parameters for both nodal and integration points. This results in a correct influence of flow in highly recirculating fiows. We do not present the details of this scheme because the results which are presented in this study are only based on the previously discussed scheme. CHAPTER 4. COMPUTATIONAL MODSLING IN TWO DIMENSIONS 109

Finite element shape fimction deridives, Section A.3, are used to txeat the pressure term in Eq.(4.78) as

where the matruc coefficients are

The role of the incompressible part of the dinosion tmis considered as an active one in deriving the integration point equation for f. Again, the elliptic nature of dinusion promotes the use of finite element shape faction derivative. After linearization of the Laplacian operator in terms of momentnm components, it can be approximated by

where Ls is an appropriate diffusion length scale, Appendix 1. This approximation redoces to fi ziS1NjFj when diffusion dominates the convection in incompressible flows, i.e. V2u=0. A general form of the Laplacian operator at an arbitrary integration point i can be written as

where the coefficient of the matrices are CHAPTER 4. COMPUTATIONAL MODELI2VG IN TWO DIMENSIONS 110

The 1st part of the discretization is the treatment of the terms inside the parenthesis in the right-hand-side of Eq.(4.?8). This part is completely lagged and compnted fiom the previous known values. The approximation of this part for an arbitrary integration point i is

With this discretization, the modeling of ail terms in Eq.(4.78) is complete. If these models, including Eqs.(4.81, 4.86, 4.89, 4.92, and 4.95), are plugged into Eq. (4.78) and similar coefficients are combined toget her the complete integration point equation for f is written as

The matrix form of the algebraic equation for f is

This integration point equation relates f at the integration point to its neighbouring nodal dependent variables. In order to derive a direct expression for f, the coefficient matrix of f is inverted and multiplied by the other coefficient matrices on the right hand side. The final form of the f operators at four integration points of the element could be summarized in the following fashion:

Comparing with Eq.(4.77), it denotes that the two last terms in Eq.(4.98) are responsible for the streamwise correction, i.e. A4. The matrices on the right hand side are named iBfluence coeflcient matrices because they ver* the correct physicd CHAPTER 4. COMPUTATIONAL MODELING IN TWO Dl2UENSIONS 111 influence of the nodal values on the integration point values. They are

{Cf) = [c'je + $fc + ,ffd]-l{~fi + ~fe) (4.101) where Cf and Cfp are two 4 x4 matrices which indicate the dect of F and P fields on f. The 4x1 array of {Cf) indudes dl known parts of the assembled terms. The first and second superscripts of C indicates to which equation and parameter of equation it respectively belongs.

Similarly, an expression for g is derived by following the procedure t hat was just described for f but starting from Eq.(4.79). The hal result can be written as

The elements of the referred matrices can be determined by comparing with the matrices of the f operator.

Now, these integration point expressions of f and g are substituted in the m* mentum parts of the arrays 7 and Ç in Eqs.(4.49, 4.50, and 4.76). According to out experience in the ondimensional study, these expressions are not plugged into the continuity equation. Supplementary integration point equations are derived in Section 4.5 for continuity equation.

4.4.2 Pressure

The integration point pressure which appeared in the conservative discretization of govaniag equations has to be calculated as a function of nodal dependent variables CHAPTER 4. COMPUTATIONAL MODELING IN TWO DLMENSIONS 112 in the element. The integration point pressure is needed for treating the pressure terms in the momentum and energy governing equations where velocity and temperature fields, respectively, are coupled with the pressure field. A strong coupling like that fiom the pressure Poisson equation, which can be derived by taking the divergence of the momentum equations, is not so critifal in this study. On the contrary, a simple interpolation will provide the primary connection and provides a good representation of the pressure field. In this regard, bilinear finiteelement shape fanctions are used to convey the eliiptic nature of pressure at the integration points

Using the following definition

will result in the matrix fonn of the integration point operator for the pressure

The minus sign in the superscript means that p is not derived based on any governing equations but on physical reasoning.

4.4.3 Temperat ure Variable

Following the procedure discussed in Section 3.4.2, the non-consetvative form of energy equation is used to derive the appropriate integration point operator for temperature. Either enthalpy or total energy as well as temperature are quantities which can be treated in the manner that convected quantities are dealt with. CHAPTER 4. COMPUTATIONAL MODELRVG IN TWO DIMENSIONS 113

On the other hand, the energy equation can be written in a nnmba of ways to represent the transport of the parameter in question, i.e. h, e, and t. Here, we select the temperature form of this equation because the temperature is considered as a dependent variable in this study. The use of other forms, i.e. enthalpy and total energy, require more linearization for h and e according to their dehi- Lions, Eqs. (2.13 and 2-14). The temperature fom of the energy equation could be formed by combining the continuity equation, Eq. (2.2), and the original form of the energy equation, Eq.(2.5). The ha1 form der employing the dehition of total energy, Eq.(2.13), assuming zero source term, and writing the convection terms in the streamwise direction, Appendix H, is

where the therrnd dissipation faction is

Comparing the l& hand side of Eq.(4.106) with that of Eq.(4.78) shows that their transient and convection terms are identical p is by k and by if replaced -PCV f t. Using this analogy and the general forms in Eqs.(4.81 and 4.86) will result in

where the matrix elements are readily obtained by CHAPTER 4. COMPUTATIONAL MODELING IN TWO DLMENSIONS 114

Contraty to Eq.(4.78), the energy dinasion term does not need linearization here,

where the matrix coefficients are given by

The pressure term is approximated by

The related matrix coefficient is written

The dissipation term is completely approximated fiom previous known values

where 5 ipi CHAPTER 4. COMPUTATIONAL MODELING IN TWO DIMENSIONS 115

Combining and rearranging the above terms WUresult in the matrix arrangement of the aigebraic equation for t as

This equation can be written in matrix form. The integration point expression for temperat ure is

The duence coefficient matrices on the right-hand-side are

Eq.(4.121) is not the only choice that can be used to calcuiate integration point temperature. Bilinear interpolation and streamwise upwinding are tno other simple methods which were applied in many cases to compute the integration point temperature. This will be discussed more in Chaptet 5 for solving mixed subsonic- supasonic flow with shock through a convergent-divergent nozzle.

4.4.4 Other Integration Point Equations

Before dosing Section 4.4, it is necessary to explain about the method of derivation of the other integration point quantities which do not directly appear in our dis- cxetized equation. One such variable is density which should provide information for deriving the velocity field from the computed momentum field at the integration CHAPTER 4. COMPUTATIONAL MODELING IN TWO DIMENSIONS 116 points. Density behaves as a convected quantity in highly compressible flows. Den- sity integration point equation could be calculated in a marner which was done in Section 3.4.2 in one-dimensional investigations. However, in tbis twdimensiond study, density was mostly computed by using the following expression

where ph- is a simple bilinear interpolation and E is a weighting coefficient which measures the co~~ectinfluence of the two approximations. It is dehed by

The two approximations return to pure upwinding and bilinear interpolating, respectively. The density at upwind point, p,, is determined in a similar manner to Eq. (4.85) for approximating f,.

The other integration point lagged dues are computed by ushg the already derived integration point expressions, i.e., Eqs.(4.98, 4.102, 4.105, 4.121, and 4.125). In this regard, velocity integration point values are obtained by using the magnitudes of the momentum components and density, i.e.,

The total energy and enthalpy integration point values are obtained by employing the new integration point values to t heir original definitions, Eqs. (2.13 and 2.14).

4.5 Mass Conserving Connections

In this section, the necessary remedy for removing the checkerboard problem which is a common problem for certain numerical methods with a colocated grid arrangement is explained. The expressions which we derived for f and g in Section 4.4.1 are CHAPTER 4. COMPUTATlONAC MODELING IN TWO DMENSIONS 117 called convected ones. As discussed before, Section 3.5, these expressions are not substituted into the control volume continuity equation. The continuity equation constrains mass through the surfaces of the control volume which is highly affected by the pressure field. Thus, the pressure field is indirectly specified by the continuity equation although there is no explicit &ect of the pressure field on density in an incompressible formulation. The use of convected expressions for f and g should compensate for the absence of a pressure term in the continuity equation. However, the investigation shows that this remedy does not necessarily temove the possibility of velocity and pressure field decoupling, e.g. Darbandi and Schneider [13], Section 3.5.1. The missing point in this condusion codd be found by reviewing the daived expressions. Convected expressions for f and g wae originally derived from the me mentuni equations. These expressions did not consider the &ect of the continuity equation which considers satisfaction of the mass. The idea of using two integration point values for a colocated grid arrangement goes badc to the work of Rhie and Chow [9]. There are ais0 other works which emphasize the employment of the two types of the integration point velocities, Schneider and Karimian [12]. Darbandi and Schneider [13] employed a new formulation for deriving a second integration point equation for momentum-components. These new expressions are obtained not only fiom modeling of the differential form of the momentum equations at each integration point, but the continuity equation is required to be used in conjunction with the momentum equations. To derive a second integration point valne for f or g, a velocity-weighted continuitysqnation error is subtracted nom the momentum governing equations. SimiIar to the one-dimensional study, the form of these equations is

(2-Momentum Eq. Error) - u (Continuity Eq. Error) = O (4.128) CHAPTER 4. COMPUTATIONAL MODELING LN TVVO DIMENSIONS 118

(y-Momentum Eq. Error) - v (Continuity Eq. Error) = O (4.129)

Nor, these equations consider the numerical mors of both the continoity and momentum equations at the integration point. For the Mke of brevity, the method of discretization is explained only for convecting f which is derived from Eq.(4.128). The results can then be extended for g. By substituthg the governing equations into Eq. (4.128) the following equation is obtained

Aii of the terms in the momentun part of this equation are treated as in Section 4.4.1. The method of discretization for the second bracket is explained here. The U% term is treated in the same manner that it was treated in Eq.(4.95), i.e.,

The two other terms are discretized ushg bilinear interpolation

4 af 8s uifi(- -lifi &fi (4.132) az + 6y j=l ipi ipi This discretization is written in matrix form for an arbitrary control volume of i

where the coefficients are defined as CHAPTER 4. COMPUTATIONAL MODELLNG aV TWO DIMENSIONS 119

There is another method for treating these two terms. In this method, is replaced by refixence to 2,with suitable alternative of Eq.(4.133), and then is discretized in streaxnwise direction. Regarding Figure 4.2, it is written

where L,+Ld, is the distance between up and dn points- fdn and f, are P~OP~~Y interpolated between adjacent nodes, Eq.(4.85). We have tested both of t hese met hods and the difference is not significant. 2 1

Figure 4.2 : Element nomenclature and veloci ty upwinding.

Now, the new derived discretizations of this section are combined with the discretization results of Section 4.4.1. This WUconclude with an equation for the convecting f , similar to Eq.(4.98) which was derived for the convected f. After substitution of the terms of Eq.(4. UO), they are categorized and combined together and the following expression is finaily obtained CHAPTER 4. COMFUTATIONAL MODELING RV TWO DIMENSIONS 120

This new integration point expression is called the convecting momentum for f. In order to distinguish it from the convected one ne denote it by j. The new iduence coefncient matricer on right-hand-side are computed by adding the new matrix derivations to Eqs. (4.994 101), i.e.,

A simila convecting could be derived for convecting j. The procedure is started from Eq. (4.129) and the convecting expression is similar to Eq. (4.M), i.e.,

{j) = [tif]{F) + [Co@]{G) + [Ch] {P)+ {CD)

These integration point expressions, i.e. Eqs.(4.137 and 4.142) are plugged into the continuity equation. Consequently, convecting velocities codd be obtained by using the following definitions, û = $ and 8 = $. Regarding the discussion in Ap pen& C and respecting the two concepts of the convected and convecting velouties, the convecting expressions are used for computing lagged values in the convection terms of the conservative form of the momentum equations, i.e. Eq.(4.36). Either convected or convecting velocities could be plugged in the lagged velocities of the energy convection terms. However, in order to respect the consistancy it is better to use convecting ones. CHAPTER 4. COMPUTATlONAL MODELING IN TWO DIMENSIONS 121

4.6 Assembly

In Section 4.3, we discretized the governing equations by nsing a mntrol-volume- based finite-element method. All discretizations were initially obtained for an arbitrary sub-control-volume and later extended for four sub-control-volumes of an element. The final results showed that the discretized equations involved nnlrnowns at integration points in addition to the main nodal unknoums, see Eqs.(4.23, 4.49, 4.50, and 4.76). This difficulty resulted in deriving expressions for dependet variables at integration points, i.e. f, g, p, and t in Eqs.(4.98, 4.102, 4.105, and 4.121). These expressions not only connect the integration point variables to theh neighbouring nodal variables but also mode1 the possible relevant physics. In addition fo f and g, supplementary integration point expressions were derived for momentum components, i.e. j and j in Ecp(4.137 and 4.142), in order to remove the pressure-velocity decoupling problem.

In this section, the discretized equations and the derived integrstion point expressions are combined and rearranged in order to provide a well-posed system of linear algebraic equations. We st art the assembly procedure with treating the continuity equation. There are two integration point variables, f and g, in Eq.(4.23). Convecting momentum components, Eqs. (4.137 and 4. M),are plugged in the continuity equation. It results in CHAPTER 4. COMPUTATIONAL MODELING LN TWO DIMENSIONS 122 where

The first and second superscripts of D indicates to which equation and parameter of equation it respectively belongs. Eq~~(4.49and 4.50) show that the 2-momentum and y-momentum equations involve pressure and momentum component unLnowns at integration points. The integration point expression for pressure is given by Eq.(4.105). The convected momentum-component operators, Eqs.(4.98 and 4.102), are select ed for unknown momentum-components at integration points. The procedure results in the following equation for the x-momentum

where

[DI = [A~~~] (4.151) CHAPTER 4. COMPUTATIONAL MODELING LN TWO DIMENSIONS 123

Similar procedure for the y-momentum equation wili resdt in

[W{F)+ [pO1{~)+ ID~I{P) + [m{T) = {PI (4.155) where

DO^] = [AO'~] (4.156)

[Dao] = [A~Q'+ AW~]+ [aoc] [Cgg] (4.157)

[Dm] = [uggc][Cm] + [am][C-p] (4.158)

[Dg'] = O (4.159)

[Dg] = {A** + AU~)- [ugoe]{Cg) (4.160)

The energy equation, Eq.(4.76), involves f, g, and t unknowns at integcation points. The unknown temperature at the integration point is substituted from Eq.(4.121) and the convecting momentum-components, Eqs.(4.137 and 4.142), are plugged into unknown moment un-components at the integration point S. It yields

[Dtt]{~)+ [Dtg]{G) + [Dtp]{~)+ [Dtt]{T) = {b) where

[D'!] = [A''' + Atfd]+ [atfc][ch] + [atgc][c~~]

[Dtg] = [A~"*+ Atgd]+ [utf '1 [cffl]+ [at#'][~ûfl]

[Dtp] = [atfe][cip] + [utge][~9+ [atk] [Cep]

[Ott] = [A'" + Attk]+ [at? [CU]

[Ot] = {A@+ Atd) - [atfc]{ci) - [atgc]{Ci) - [atk]{Ct) CHAPTER 4. COMPUTATIONAL MODELING IN TWO DIMENSIONS 124

The first stage of the assembly is finished now. We have already derived well- posed discretized equations for the continuity, Eq.(4.143), z-momentum, Eq. (4.149), y-momentum, Eq. (4. and energy, Eq. (4.161), equations. Each of these equations consist of four sub-equations for four sub-control-volume of an element. The number of nnlniowns in each sub-equation can be 16 or less. This number indicates the m&um number of unknowns at four nodes of element. They are four Fs, four Gs, four Ps, and four Ts.

In the second stage of the assembly, the derived elementai equations are put toget her

[at]{F) + [Dgg]{G) + [DOP]{P) + [Dg']{T) = {Dg) (4.167~)

[D"] {F)+ [Dîg]{G)+ [Dtp]{P)+ [Dtt]{T) = {Dt) (4.167d)

It could be written in the following matrix form as

This 16 x 16 matrix is named elernental sti'ness matriz. As seen, the procedure of the assembly resulted in a well-posed system of algebraic equations having 16 equations and 16 unknowns. It is important to note that this mat^ does not provide information related to the conservation of the quaatities because the assembly pr* cedure is done for four sub-control-volumes of four difkent control-volumes. While CHAPTER 4. COMPUTATIONAL MODELLNG IN TWO DIMENSIONS 125 the governing equations are consened within individual eontrol volumes rather than element s or sub-control-volumes .

At the third stage of assembly, the elemental stiffhess matrix is assembled into the global matrix of alI dements. In this regard, each derived snb-equation is added to the related global conservation equation. This global equation belongs to the control volume that the derived sub-equation belongs to one of its four sub-control- volumes. The assembly of four different sub-control-volumes of a control volume will result in the full conservation of the conserved quantities for that control volume. This is one of the most important advantages of the control-volume-based methods which provides the conservation laws for finite volumes.

The resdted system of algebraic equations has been solved by a direct sparse matrix solver, Chu et al [7O].

4.7 Boundary Condit ion Implementation

Before presenting the results of the application of the numerical method, it is important to describe the techniques used to invoke boundary conditions. This not only eliminates repeating these discussions and thereby saving space but it also provides a collection of wor t hwhile detailed information for future rekence.

The assembly of the elemental equations, Section 4.6, results in complete conservation of mas, momentum, and energy equations for all control volumes of the domain except those which have one or more surfaces coincident with solution domain boundary. The process of dosing the conservation equations for a boundary control volume is completed if mass, momentum, and energy boundary flows are CHAPTER 4. COMPUTATfONAt MODELING IN TWû DIMENSIONS 126 taken into account, i.e.

(interna1 z-moment- equation)+ Q[ = O (4.170)

(internal y-momentum equation)+ QI= O (4.171)

(interna1 energy equation)+ Qi= O (4.172) where Qbls represent the boundary mass, x-momentum, y-momantum, and energy 0ows which are indicated by m, f,g, and e, respectively. They consist of elementary convection and diffusion flux terms

These fluxes are calculated by approximating the integration over the sudaces which are coincident with the domain boundary surfaces. These surfaces have been illustrated for the control volume in question in Figure 4.3. The crosses at the surface mid-points represent them, i.e. at bipl and bip2 integration points. The resulting flwes for the continuity equation are writ ten CHAPTER 4. COMPUTATIONAL MODELING IN TWO DIlMENSIONS 127

Similady, the resulting fluxes for the momentum equations are given by

And for the energy equation, they are are written

where a, and ut are normal and tangential surface stresses. (AS), and (AS), were dhedbefore by Eqs.(2.25).

A comprehensive discussion on the proper treatment of the above expressions and their combinations at bonndary surfaces has been presented by Schneider [71]. CHAPTER 4. COMPUTATIONAL MODELING IN TWO DIMENSIONS 128

volume

I Figure 4.3: Boundary control volume and boundary condition implementation.

There are many dinerent methods to implement the boundary conditions. How- ever, we are not interested in presenting all possible boundary implementations; conversely, we are only concenied with those implementations which have been employed in the test problems of Chapter 5.

In the current study, the diffusion parts of the boundary conservation equations, i.e. Qbfd ,Qb @d , and Qgd, are treated assuming zero viscous diffusion through the domain boundaries. The value of all other Qb9sare approximated by linear in- t erpolation between adjacent boundary nodes unless the conservation equations are sacrificed to spedy the known boundary values or their directions. For example, Equation (4.178) could be expanded for the control volume in question in Figure 4.3,

Genady speaking, the mass and momentum equations are used to tteat pressure and velocity boundary conditions and energy is involved in treating tempera- CHAPTER 4. COMPUTATlONAL MODELlNG IN TWO DIMENSIONS 129 ture boundary conditions. The main purpose of the present boundary treatment is to conserve mass for all control volumes of the domain including the boundary. This is why the known pressure at the idow and outflow boundaries of subsonic flow domains is speded by nsing the momentum equations. Two factors are important in boundary treatment. One is the nature of the flow and whether it is viscous or inviscid. The other relates to the physics of the boundary which muld be classsed into solid wall, iaflow, outflow, etc. Based on these factors and sub-factors, several different boundary treatments could be constmcted whidi are explained in the following sub-sections.

Ncdp boundary conditions are applied to mode1 solid walls for solving the Navier- Stokes equations. In this regard, the 2-momentum and pmomentum equations are replaced by F=O and G=O specifications for stationary wab and F=Fwu and G=Gdl for moving walls. Continuity is automatically closed for solid boundary becanse there is no mass flow through the solid boundary. Similady, energy is not transmitted from boundary surfaces because boundaty velocities are zero. Thus, an 8T adiabatic boundary condition is applied to the energy equation, i.e. q, = k-~n = 0.

Inviscid Solid Wall Modeling

The flow tangency condition is considered for treating inviscid Euler flow over a solid wd. This condition restricts the direction of the flow to be tangent to the wall boundary, Figure 4.3. In this case, the 2-momentum and y-momentum equations are completed normdy for boundary control volumes assuming zero vismus CHAPTER 4. COUPUTATIONAL MODELLNG IN !IWO DIMENSIONS 130

diffusion. Then, they are combined to form the tangential momentum equation,

ms(@) {z-moment* + sin(p) {y-moment-) = O (4.187)

Now, the x-momentum equation is replaced by this new derived tangential momentum equation for snch boundary control voIumes. The y-momentum equation is simply replaced by the normal n-flow condition for the boundary control volumes,

Since there is no mass flow across the inviscid solid wall, continuity and energy equations are treated similar to those for a viscous solid wall.

Inflow Boundary Modeling

Most flow information is specified at an inlet section. In all subsonic, transonic, and incompressible cases, the x-component of mass or velocity is specified by replacing the 2-momentum equation by an equation that simply specifies the mass comp* nent or x-velocity. The y-component of mass or velocity is not always specified. For those test cases which are consistent with a uniform inlet velocity dis tribution, the y-momentum equation is discarded for corresponding boundary control volumes and replaced by G=O instead. However, for those inlets with non-dom inflow, like the sloped inlet of a convergent-divergent nozzle, the y-moment um equation is completed normally considering zero shear for those control volumes. The result of t his boundary implement ation is illustrated and mentioned in applications. Con- trary to the momentum equations, the continuity equation is always dosed for the above flow cases.

A dorm inlet velocity specification does not necessarily describe a uniform mass distribution at the inlet and vice versa in compressible flow. In many cases, CHAPTER 4. COMPUTATIONAL MODELING IN TWO DIMENSlONSIONS 131

it is necessary to compare our results with those of other workers who solve using a veiocity boundary condition. In order to have a oniform velocity at an x-inlet for our momentun variable procedure, F is specified via the x-momentum equation. Then, density is computed by using the equation of state and having the magnitude correspondhg to the calcdated pressure and temperature. This calcuiated density is used to correct the inlet mass, FM, = @-CI,,.This new derived F is used to specify mass in the next iteration. Although this method of boundary implementation generally slows down convergence, it works well.

For supersonic cases, P, F, and G are specified at the inlet by replacing the continuity, x-momentum, and y-momentum equations, respectively. Temperature is always specified by replacing the energy equation by an equation that simply specifies the temperature. This latter implementation is applied for all investigated tests.

Outflow Boundary Modeling

At an outflow boundary in the z-direction, neither the x-velocity nor x-component of the mas is specified. Pressure is speaed here by replacing the 2-momentum equation with an equation specifying the desired pressure for subsonic and transonic flows. The method of treating the y-momentum equation is exactly similar to the one presented in inflow boundary modeling. The continuity equation is closed by an implicit treatment of the exit mass for subsonic and transonic flows.

Since pressure is speeified at an upstream boundary in supersonic flow (see inflow boundary modeling) x-momentum equation is dosed assuming zero shear. There are few test problems in supersonic flow which speofy either pressure or mass at outflow by extrapolating specified values fiom the upstream of boundary nodal CHAPTER 4. COMPUTATIONAL MODELING IN TWO DllMENSIONS 132 points and inside of the domain. However, this form of modeling does not cause signiticant improvement in the result S.

DSerent techniques are osed to deal with the energy equation for outflow boundary control volumes. In most cases, the energy equation is completed subject only to zero shear. In this modeling, dependent variables are implicitly involved in closing boundary values. In supersonic flows, the energy equation is 4th- completed as before or temperature spedied with magnitude extrapolated fiom neighborhg upstream nodes. Another technique is to use an adiabatic boundary condition, q, = ks= O. Such an equation relates each boundary node to its neighbonring nodes using fmite elernent connectors or a direct badrward dinerencing.

Symmetry Boundary Modeling

Symmetry boundary conditions are very beneficial for redueing cornputer theand memory requirements when appropriate. The mas, momentum, and enagy flows do not intersect the symmetry boundary surface because the normal velocity and normal derivative of all other flow parameters are zero on that surface. This M- plements a normal dosing of the continuity equation. In this modeling, the z- momentum is replaced by an equation which specifies zero gradient for tangentid velocity at the boundary, i.e. = O. The y-moment- is aiso replaced by an equation, Eq.(4.188), which spedes the direction of the symmetry boundary. The adiabatic boundary condition is implemented for treating temperat are as was done for viscous solid wall modeliag. CHAPTER 4. COMPIITATIONAL MODELING IN TWO DIMENSIONS 133

4.8 Closure

The preliminary investigation on one-dimensional formulations has been extended in t his chapter for twdimensional application. In t his regard, the twdimensional Navier-S t okes equations were discretized using a control-volume-based finite- element method and selecting the momentnm component variables as the dependent variables. The discretized form of conservation equations contained many nonlinearities and formed an ill-posed systern of equations due to the integration point variables involved. Both of these difficulties were resolved using our one-dimensional experîence. The velocity-pressure decoupling issue which was detected in the one- dimensional investigation was addressed folowing our previous methodology. The resulting equations in this chapter were cast in a general matrix tom which will fa- ditate the future work. The details of the techniques which are employed to invoke boundary conditions were explained for different bonndary condition implementation. The performance of this formulation will be studied and results presented in the next chapter. Chapter 5

Two-Dimensional Result s

Introduction

In this chapter, the performance of the developed two-dimensional method is examined. This examination comprises the investigation of several difFerent kinds of flow and test problem. The main purpose of this chapta is to cover a wide range of flows and problems. Based on the title of this research, the primary concern in this route is to test the validity of the method for two major types of flow, i.e. incompressible and compressible flow. There are 0th- factors like speed, viscosity, and tirne which break each type of flow into other categories. The time variable splits problems into steady and unsteady flows and viscosity into viscous and inviscid flows. Flow speed is a major parameter in compressible flows which divides it into many other subdinsions like subsonic and supersonic flows. Flow problems can be huther categorized based on the flow boundary conditions.

The one-dimensional resdts were completely discussed in Section 3.7. Here, we are concemed only with twdirnensional flows and mainly with interna1 flows. In CHAPTER 5. TWO-DIMENSIONAL RESULTS 135 order to MiIl the objectives of this chapter, the flow problems are dassiîied into three categories of incompressible, pseudcxompressible, and highly compressible flows. The hst is limited to absolute constant density flows, the second addresses incompressible flows which are treated as compressible ones, and the last addresses highly compressible flows. Each component has specially been designed in such a manna that their assemblage would demonstrate the power of the method for a broad range of flows. Many other types of flow are examined in this chapter including viscous and inviscid flows. The factor of time ras examined in Section 3.7.2 where the one-dimensional transient flow in a shock-tube was investigated.

Many diffkrent problems are modeled in this chapter induding the driven cavity problem, channel entrance flow, converging-diverging nozzle flow, and flow over both a bump and a ramp. There are three major examples which are tested almost in ail three parts. They are the cavity, entrance, and nozzle flows. This helps to follow the performance of the method through a wide range of fiow parameters. The general common specifications of each sub-section is generally discussed before presenting the individual results of that sub-section.

In this chapter, the results of the present work are compared with those of other workers. Their results have been extracted either fiom the figures of their articles by using a digitizer or dicectly fkom the tables of their articles. These results may simply be indicated by symbols instead of continuous line distributions. The number and location of the symbols do not necessasily indicate the grid distributions.

In this chapter, all test problems are solved based on a tirne-dependent impliut algorithm. There are no interna1 iterations in each the step because we are concerned ody with the steady-state solution. Marching in tirne is continued until a convergence criterion is reached. The Root Mean Square of the change fiom iteration (tirne step) to iteration is used as a measure of convergence for the dependent CHAPTER 5. TWO-DIMENSIONAL RESULTS 136 variables, i.e. F, G, P, and T. For a general unknown #, the RMS is given by

where

H is the total number of grid nodes. The RMS is evaluated for all four unknown dependent variables, however, experience shows that the RMS of F is slower than the other variables for the test cases investigated . The RMS values for the dependent variables at the integration points are also checked using similar defmitions, however, experience shows that the convergence of the integration point variables is faster than t hat for the nodal values. The required criterion for convergence was set to IO-' for all cases unless otherwise stated.

AU pre and post processing of this chapter have been accomplished using either MATLAB software or author based tools. W figures which illustrate the results have been depicted by MATLAB. Since the intent of this work was to demonstrate the procedure, a direct sparse mat& solver was used for the computations [70]. Thus it is not relevant to present computational times for solutions in this work.

5.2 Incompressible Flow

As the first step of our method validation, the capabilities of the incompressible algorithm of the code are investigated. In this algorith, the equation of state is reduced to Eq. (2.16) where absolute incompressible flow is spedied. In this section, two test models are examined to reveal the characteristic of the method. They are CHAPTER 5. TWO-DIMENSIONAL RESULTS 137 the &en cavity problem and the entrance flow problem. They are addressed by solving the Na+-Stokes equations. Since viscous flow problems may be polluted by turbulence &ects it is necessary to respect laminai. restrictions for laminar flow solvers. The validity of the results is compared with the results of other numerical methods and available benchmark result S. A mes h sefinement st udy is accomplished for the cavity test problem which demonstrates the ability of the method to attain accurate results for coarse grid distributions. All test cases in this section are solved within a time-marching algorithm and the results are obtained by choosing huge time steps which results in an infinite CFL number.

5.2.1 Driven Cavity Problem

The fust model problem is a classical problem to test the accuracy of numerical methods for incompressible viscous flows. It is the recirculating flow of a fluid in a square cavity which is bounded on three sides and whose fourth side moves at a constant speed, causing recirculation inside the cavity. The cavity problem is a difncult test problem because of two singularities at the corners of the lid and because of its several recirculation regions with their complexities dependent on the Reynolds number. This problem has been extensively studied as a benchmark problem, Chia et al [72]. In order to demonstrate the etof mesh size and Reynolds number on the results of the code, the study was done on five different uniform meshes, including 21x21, 31x31, 41x41, 51x51, and 71x71 grid nodes, and for three Reynolds numbers of 1000, 5000, and 7500. The length scale and density were considered to be unity. All velocities are nondimensionalized by the velody of the lid. Figures 5.1, 5.3, and 5.5 depict the streamline contours for the tkm Reynolds number of 1000, 5000, and 7500 on three different udorm grids of 31x31, 51x51, and 71 x71, respectively. Plotting the streamline contours requires CHAPTER 5. TWO-DIMENSION.RESULTS 138 the evaluation of the stream function within the solution domain. In this regard, the stream function values have been obtained by integrating the integration point velocities over control-volume surfaces. Al1 the first and second level vortices have been successfully detected in these three figures despite using relatively coarse grids. As Reynolds number increases the central vortex becomes much roundex and the secondary vortices much stronger.

The u-velocity and Y-velocity profiles at the centerlines of each cavity have been calculated and iliustrated in Figures 5.2, 5.4, and 5.6 for three Reynolds number of 1000, 5000, and 7500, respectively. In order to study the effect of mesh refinement on the accuracy of the results, each Reynolds number was tested on up to three dinerent grid distributions and compared with the benchmark results of Ghia et d [72] whîch are based on fme grid distributions. Their working grid is 129 x 129 for Reynolds 1000 and 257 x 257 for two other Reynolds numbers. Comparing with the results of benchmark work, this study shows that the accuracy of the solution is r apidly increased wi t h relat idy coarse grids .

Following the mesh refinement study, a comparative study on the validity of the method is performed. Reynolds number of 3200 is selected for this study because of adable results. Figure 5.7 compares the results of the present method with three other teferences two of wbich are based on d-speed methods. The benchmark work [72], which has already been introduced, uses a grid resolution of 129x 129. The grid resolution is 71 x 71 in (451 and 51x5l in (331. The present method hap used a grid distribution of 51 x51 which generdy shows better agreement cornparhg with the benchmark resuits. The streamline contour of this cavity fiow is presented in Figure 5.8.

Certain cavity flow details have been studied and compared with those of the benchmark study. Table 5.1 shows a comprehensive survey of the primary and sec- CHAPTER 5. TWO-DIMENSION'RESULTS

Figure 5.1: Cavity with a 31 x31 grid and Re =1000.

grid study on cavity (Re=1000) U distribution on the vertical center line

C O ---3 0.4

Ho.,)i - -0.5 O 0.5 1 u velocity V distribution on the horizontal center line

x position of nodes

Figure 5.2: Mesh rehement study on centerline veloaties. CHAPTER 5. TWO-DIMENSION' RESULTS

Figure 5.3: Cavity with a 51x51 grid and Re =5000.

grid study on cavity (Re=5000) U distribution on the vertical center line

V distribution on the horizontal center lin0 1-1 0.6 1-1

O 0.5 1 x position of nodes

Figure 5.4: Mesh rehement study on centerlhe velocities. CHAPTER 5. TWO-DIMENSIONAL RESUCTS

Figure 5.5: Cavity with a 71 x7l grid and Re =7500.

grid study on cavity (Re=7SOO) U distribution on the vertical center lin8

4.5 O 0.5 1 u velocity V distribution on the horizontal center line

O 0.5 1 x position of nodes

Figure 5.6: Mesh refinement st udy oa centerline velocities. CHAPTER 5. TWO-DIMENSIONAL RESULTS

U disuibution on the vertical center line

u velocity

V distribution on the horizontal center lin0

- Present work 51 x51 O Ghia et al 129x129 @2] + Chen 6 Pletcher ilxi1 (451 x Karimian 8 Schneider 5 1x51(33]

O 0.5 1 x position of nodes

Figure 5.7: Comparative stndy on the centerline velocities in cavity, Re=32OO. CHAPTER 5. TWO-DIMENSIONAL RESULTS

maln Gnes

Figure 5.8: Cavity with a 51x5l grid and Re =3200. ondary vortices of the flow inside the driven cavity for Reynolds number of 5000. This survey includes the strengt hs, dimensions, and locations of the main vortex and secondary vortices in the cavity. In this table, several abbreviations are used. P, T, B L, and BR represent primary, top, bot tom left, and bot tom right vortices. H and V represent the horizontal and vertical sizes of the vortex respectively mea- sured dong the correspondhg wall surface. The summarizeed results in Table 5.1 demonstrate good quantitative agreement with the redts of Ghia et al [72]. It must be noted that the grid size has a ciirect and strong dect on the order of errors and that vortices of smailer size would res& in higher errors on coarser grids. The smaiier vortices have not been presented in this table. SMarcornparisons for other Reynolds numbers show similar results.

Ndip boundary conditions were appiied on all soiid walls of the cavity. The cavity problem was studied as a steady-state problem and the results of each case were obtained using several iterations in a huge time step. The nomber of itera- Table 5.1: Comprehensive cornparison of cavity details for Re =5000

tions was between 10 and 20 in each case to reach the reqnired criterion which is RMS<~O'~for F and G.

5.2.2 Channel Entrance Flow

The second test problem is entrance flow between pacallel plates. The parailel plates geometry is a limiting geometry for the fdyof both rectangular and con- centric annular ducts. The velocity distribution at the inlet of a duct wilî undergo a development from some initial profile at the entrance to a Mydeveloped profile at locations far downstream. The region of the duct in which this velocity develop ment occurs is called the entrunce region There has been considerable interest in determining the fluid behaviour Mthin the entrance region because of its general technical importance in engineering applications. The importance of this problem is also for developing laminar-flow theory and testing numerical sdemes for solving ellip tic conservation equations. There are no general exact solutions or experimen- td results for the entrance region. However, there are a variety of approximate analy tical and numerical met hods for the determination of the flow characteristics in this region, Darbandi and Schneider [73]. The development of a laminar flow at the entrance of two semi-infinite straight parallel plates is seen in Figure 5.9. The distance between two plates is H = 1. The entrance length, Xe,is defined as the

y------t & Entrance Length

Figure 5.9: Developing and developed zones. distance from the inlet boundary, with dominlet velocity profile, to the point where the centerline velocity reaches 99% of its asymptotic value. The entrance length divides the region into two zones. In the developing zone the velocity profile undergoes a transition from a flat to a parabolic profile. This developing profile may have two maxima at locations other than the centerline. The patabolic profile remains constant in the fdy developed zone. Ali lengths are nondimensionalized by H and velocities by U'det. Because of the symmetrical nature of this problem, only the upper or lower half-channe1 could be calculated. However, in order to CHAPTER 5. TWO-DIMENSIONAL RESULTS 146 emphasize on the symmetry of the obtained solution, the calculation is done based on a full height of the channel.

The problem ras investigated for Reynolds nnmbas O, 1, 20, 200, 1000, and 2000. The fitst one is creeping flow and is approximated by specifying a very large value for viscosity. A 101x41 grid distribution was used for all Reynolds numbers. The fist and second numbers show the longitudinal and the fdtransverse grid distributions, respectively. The distribut ion dong the cross section was based on the hyperboiîc sine and the longitudinal distribution was uniform.

Typical profiles of u-velocity for Reynolds numbers 200, 1000, and 2000 are shown in Figures 5.10 to 5.13 assuming a domvelocity profile at the inlet. Qualitatively, the development of the velocity profiles were found to be quite similar at all Reynolds numbers although the off-centerline maxima appear to be the greatest at Re=200. As is seen, the velocity profiles have a peculiar behaviour close to the entrance. They show a local minimum at the center of the duct and symmetrically two maxima near the wds. These velocity overshoots are found at ail Reynolds numbers while their magnitudes deuease and finally vanish with increasing Reynolds number. Shah and London [74] present a comprehensive discussion on the existence of these overshoots. They resdt iiom the condition that the velocity distribution at the inlet must be uniform. In order to maintain this condition, an adverse pressure gradient develops in a smd region on the centerline near the entrance. Fluid parcels near the centerline are not accelerated immediately where as the 0uid parcels next to the wdare forced to be stationary as soon as they enter the inlet region. To satisfy the continuity equation, velocity overshoots are t bus forrned.

There have been numerical [75], analytical [76,77], and experirnental[78] dorts to determine whetha or not these overshoots are a part of the real behaviour of the CHAPTER 5. TWO-DIMENSIONAL RESULTS

Figure 5.10: Entrance flow, Re=200, undeveloped.

profile of u-mbQty in difieren t posilians in channu4

Figure 5.11: Entrance flow, Re=200, developed.

profile of u-veiocity in different positions in channd

Figure 5.12: Entrance flow , Re=1000, undeveloped.

profile of u-vdodty in different positions in channd

Figure 5.13: Entrance flow , Re=2000, undeveloped. CHAPTER 5. TWO-DIMENSIONAL RESULTS 148 flow. Abarbanel[76] analytically solved the problem for Stokes flow and conduded that these bulges are indeed a real part of the mathematical solution. Berman and Santos [79] demonstrated wit h t heir experimental work that these velocity overshoots in the entrance region for pipe flow are not just a mathematical oddity but are real [8O]. On the other hand, in the numericd category, AbddNour and Pot ter [SI] have solved the entrance flow of ducts by applying both dorm and actual inlet profiles and using vorticity and stream-hction variables. They concluded that both improvement in the boundary conditions and the use of smoothing functions could rninimize the magnitude of these overshoots. Darbandi and Schneider [73] have conducted a comprehensive study on the effect of mesh-rehement on velocity overshoot in the entrance region of a channel flow. They conduded that the velocit y-over shoot distribution dong the channel approaches asymp totic values with mesh rehementS.

In order to study the effect of mesh size, the mesh size was changed from a non-uniform 101 x 41 grid to a 61 x 21 uniforxn grid distribution. The effect was low on the results of entrance length computation. Their changes were consistent with the change of mesh size at the end of developing region. The entrance length ha9 been nondimensionalized with H and tabulated in Table 5.2 for different Reynolds numbers. The results of the present work are compared with those of Narang and Krishnamoorthy [82] who solved the boundary-laya equations and Morihara and Cheng [75] who solved the quasi-linear Navier-Stokes equations for incompressible flow. Although these results are based on the her grid but they are not far fiom the results of the coarse grid which infêrs excellent result despite the use of a coarse grid. N.A. 1.282 2.220 16.70 91.080 168.80

Table 5.2: Entrance or developing flow length

There is an empirical relationship for the hydrodynamic entrance length,

This expression is not valid for Reynolds numbers under 100 because Lhv is a strong function of Re for low Re flows. This length has been calculated for ou. test problem and has ben tabulated and compared with others' results in Table 5.3. The number of iterations for attaining the convergence criterion, i.e. RMS<10-5 for F and G, did not exceed six itaations, for the highest Reynolds number, in all test cases.

Boundary Conditions

The Navier-Stokes equations for pardel plates may be solved for different inlet conditions. The selection mainly depends on the dependent variables. Those who solve for cpC use irrotational entry conditions, lEC, at the inlet and those who solve using u-v use uniform entry condition, UEC. There is a third option which C'TER5. TWO-DIMENSIONAL RESUZTS

Re Present Morihara [75] AbddNouz [81] empirical result 20 0.0558 0.0559 0.045 0.04 200 0.0456 0.0452 0.0442 0.04 1000 0.0441 N.A. O .O442 0.04 2000 0.0436 0.0429 0.0443 0.04

Table 5.3: The hydrodynamic entrance length

considas the dom-velocity fat upstream of inlet section, e.g. (831, which is not considaed here. Van Dyke [84] pointed out that the vorticity at the inlet is not zero for low or moderate Reynolds numbers because of its upstream diffusion as soon as the flow mets the entrance wall. Hence, the unifolpi entry velocity distribution is better than a zero vorticity distribution. Morihara [75] calculates and plots equi- vorticity lines in the duct and shows that the rp-C formulation is justified ody for large Reynolds numbers .

There are works which apply both boundary conditions. Mcdonald [85] solved the complete set of Navier-S tokes equations for both the domand irrot ational entries. He noted that the centerline velocities are higher for the irrotational entry than for dormentry. This difference is higher for parallel plates compared to the circular tube. Ramos and Winowich [86] investigated magnetohydrodynamic channd flows. They showed t hat the primitive-variable formulation predic ts either steeper axial velocity gradients at the channel walls or lower tucial velocities at the Channel centerline than the stream fimetion-vorticity in fiaite-dinesence or finite- element methods. On the other hands, AbddNour and Potter [SI] show that the magnitude of overshwts could be minimized with improvement of boundary conditions in a y+( formulation. Ail these cornparisons between tr-v and (p( formulations show that thewould be tao different solutions considering either UEC or IEC. These solutions ate not identical.

In this study the initial solution is always started with F=G=O aad P=Po at all mesh nodes throughout the domain. Then dormen- condition, i.e. F=l and G=O, is applied at the inlet. Almost all of the methods in the literature assume fdy developed flow at infinity. Here, the dormpressure and zero transverse velouty are speafied far downstream of the entrance length, Xe, for boundary condition implementation. Since the problem is solved for the haIf height of the domain, symmetnc boundary conditions are applied at the centerline. In this regard, mass flux and its related momentum are considered zero through the centerline. No-slip conditions are spefied on the wdof the duct.

5.3 Pseudo-Compressible Flows

In this section, we are mainly concerned with the performance of the analogy to employ a compressible algorithm to solve for incompressible flows. Therefore, we are directly interested in evalaating the ability of the code to handle low-Mach- number flows known as pseud~compressibleflows. These flows definitely have the characteristics of real incompressible flow. In order to demonstrate the pdormance of the analogy, many pseudcxompressible flows are first solved using the compressible algorithm of the code, using Eq.(2.11) as the equation of state, and the results are compared with the results of the incompressible algorithm of the code, using Eq. (2.16).

In the following sections, there will appear a minimum low Mach number in each test case but this does not mean they are the lowest possible Mach numbers which could be solved by the compressible algorithm. It is the Mach number which definitely reveals the characteristic of teal incompressible flow. Cornparhg the resdts of these sections shows that there is no difference between these resnlts and those of incompressible flow although deasity was permitted to obey the ideal gas law. This high flexibility of the method in solving real incompressible flow as a compressible flow is not seen in compressible methods which are extended to solve incompressible flows. Volpe [30] has examined the performance of three twe dimensional compressible flow codes at low Mach nunibers which did not exhibit this flexibility.

In this section, the performance of the analogy is investigated in three different test cases of cavity problem, entrance flow problem, and converging-diverging nozzle flow problem. The two hst are solved by treating the Navier-S tokes equations and the last using the Euler equations. In order to complete the investigation, all compressible and pseudo-compressible results are compared with either analytical solutions or benchmark results.

5.3.1 Cavity Flow Problem

The fist mode1 problem is the two-dimensional cavity àriven by the movement of the lid. The complexity in flow conditions is associated with well dehed boundary conditions. This test problem was introduced in Section 5.2.1. The ability of the curent work in detecting the separate recirculating regions of the incompressible cavity was previously demonstrated by solving high Reynolds number cavity flows, Section 5.2.1. Here, we are not directly concerned with the details of the solution but the abiüty of the analogy to employ compressible algorithms to solve for incompressible flows. Following this purpose, two cavity problems with grids of 19 x 19 and 31x31 are selected to study tao Reynolds nnmbas of 100 and 1000, respectively. The cavity has a unit length scale and velocities are nondimensionalized by the lid velocity.

These problems are solved once by the incompressible algorithm and several times by the compressible algorithm. In the compressible case five difFerent Mach numbers are investigated. These Mach numbers are divided into hro degories of pseud~compressible,M ~0.3,and compressible, M >0.3, Mach numbers. Again, it must be noted that the lowest Mach nnmber of M=0.00001 does not mean that it is the lowest possible Mach number for using the compressible algorithm. Figure 5.14 illustrates the u-velouty profiles at the vertical centerline of the cavity and Figure 5.15 similarly does for the v-velocity profiles at the horizontal centerline. These figures demonstrate the results of both absolute incompressible flow and compressible flows fiom very low Mach numbers up to sonic speed. These velocities have been nondimensionalized and compared with the incompressible results of Ghia et al [72]. As these two figures show all results are absolutdy identical for both compressible and incompressible algorithms. There remains a question whether they are obtained under dinerent solution conditions. Surprisingly, the answer is negative, i.e., ail side conditions are definitely identical. Figure 5.16 compares the convergence history of F for diffaent cases. As seen they are dl identical. This is a feature not seen in the conventional compressible flow solvas and even in theif modified versions which solve for incompressible flows.

One important issue which needs to be qlained here, is the velocity profiles in highly compressible flows. The cornparison shows that the velocity distributions are identical dong cavity centalines for both highly wmpressible and rdinwmpress- ible flows. The reason behind this similarity retums to the velocity distribution inside the incompressible cavity. As Figures 5.14 and 5.15 show, most of the region inside the cavity is under incompressible conditions, i.e. Mach < 0.3. This CHAPTER 5. TWO-DIMENSIONAL RESULTS

U distribution on the vertical center Iine

Figure 5.14: The velocity distributions for vertical center grid of the cavity problem,

V distribution on the vertical center lin8 oz1 I

x position of nodes

Figure 5.15: The velocity distributions for horizontal cent- grid of the cavity problem, Re=100. Convergence History of F

WMsdiNo. x x Mm - MtO.40 , -.-.- w.1 - M4.01 ...... ~.oOOol + + InoanpeasBb

Figure 5.16: Cornparison of the convergence histones in cavity with Re=100. causes a sharp drop of compressibility efTects beside the cavity lid. On the other band, the cavity region is almost a constant pressure field except at the regions dose to the leading and trailing edges of the moving lid. This has a direct effect in generating a constant density field. This independence of the solution fiom Mach number ha9 also been reported by Pletcher and Cheng [38]. This is why we have included the results of high subsonic flows with those of pseudc+compressible flows. In another words, the behaviour of high Mach nnmber subsonic flow is similar to that of very low Mach number flows and they could really be categorized in the pseud~compressiblebranch.

The convergence study of the analogy is compared with that of Pletcher and Cheng [38] over a range of Mach nambers in Table 5.4. The cornparison is performed for both preconditioning and ncxonàitioning procedures. In this table NO1 stands for the Number-of-Iterations and NA means the case is Not-Available. For the Precondi tioning [38] This Work

Mach 10-~ IO-= IO-^ o. 1 0.2 0.4 0.8 1.0

Table 5.4: Cornparison of the results of different compressible schemes for cavity flow, Re = 100, grid 19x 19 ncxonditioning scheme, it is not possible to use the same time step over a wide range of Mach numbers. They also could not reduce the number of iterations for the no-conditioning scheme to the level achieved with preconditioning for Mach numbers lower than 0.8. For the current study, the convergence was determined for the steady calculations when the Root-Meamsquare of ali dependent variables reached IO-'. The stability of the resdts of the curent analogy in achieving the convergence char acteristics are excellent in cornparison wit h t hose of the refkrence. The low number of time steps is another issue which demonstrates the ability of this work. This is not seen in conventional compressible methods which are applied to very low speed flows. Volpe [30] examines three different Euler and Navier-Stokes solvers at different low Mach numbers and deduces that the number of iteration cycles to reach the convergence criterion is excessively high.

Many similar caldations were accomplished for other Reynolds numbers and grid resolutions and the similar results were achieved. As an example, Figures 5.17 to 5.19 demonstrate a similar investigation and cornparison for a higher Reynolds number, Re=1000, in the same cavity. The mesh size is 31 ~31in thh model. In order to illustrate the trend of convergence for lower RMS aiterion, this criterion has been diminished to 10-~.The secondary reeirenlsting regions become stronger

Convergence History of F

Figure 5.17: Cornparison of the convergence histories in cavity with Re=1000. at this Reynolds number. Despite this higher complenty of the flow, a similar conclusion which was derived for Re=100 is once more determined here for Re=1000. It is noted that al1 identical results show identical rates of convergence. Section 5.4.1 present s more result s for compressible cavity with higher Reynolds numbers. CHAPTER 5. TWO-DIMENSIONAL RESWLTS

U distribution on the vertical center line

Ü / Ulid

Figure 5.18: The velocity distributions for vertical center grid of the cavity problem, Re=1000. V distribution on the vertical center lin8

Figure 5.19: The veiocity distributions for horizontal center grid of the cavity problem, Re=1000. 5.3.2 Channel Entrance Flow

The schematic development of a laminar flow at the entrance of hosemi-infinite straight parallel plates was shown in Figures 5.10 to 5.13. This mode1 problem has been inves tigat ed by the incompressible algorithm of the momentum-component procedure in Ref. (551 and Section 5.2.2. It has been shown that the velocity profile within the developing zone may have two maxima at locations other than the centerline. Here we are not directly interested in the overshoots or their magnitudes but in the performance of the analogy in solving both compressible and incompressible entrance flows.

Figure 5.20 illus trat es the centerline velou ties for incompressible flow and four compressible flows with inlet Mach numbers of 0.001, 0.01, 0.05, and 0.1 and a grid distribution of 41x21. As is seen, compressibility dects are not important for M 50.01 and the results are identical with those for incompressible results. However , compressibiüty effects become noticeable as the inlet Mach numba ap proaches 0.1. The derived results are compared with those of Morihara and Cheng [75] who solve the quasi-linear Navier-S tokes equations for incompressible flow . In addition, they are compared with the compressible results of Chen and Pletcher [45] at M=0.05. Generally speaking, the agreement between the results is excellent. Figure 5.21 compares the convergence histories of the investigated cases. The low number of tirne steps to achieve the criterion of for d dependent variables is excellent. The deviation for M=0.1 is expected due to compressibility effects and the process of speeifying the velocity at the inlet. The latter means that at the end of each time st ep the masses at inlet nodes are corrected by multiplying the specified valaes of velocity and the new calculated densities. These new masses are employed as speded masses at the inlet for the next time step. When compressibility efEects CHAPTER 5. TWO-DIMENSIONAL RESULTS

U distribution on œnterline grid 1.6 . O oMomioro&Chengp5l

lnbt Mech No. ,.,,. Mr0.001 -MIO.01

-.-*- M-0.05 -M-0.1 + + kicompressble

Figure 5.20: Centerline velocity distributions for incompressible and compressible flows in entrance region, Re=20.

Convergence History of F 1O"

Figure 5.21: Comparing the convergence histories for entrance flow , Re=20. CHAPTER 5. TWO-DIMENSION' RESULTS become noticeable, this correction slows the convergence rate.

A similar presentation is given for Re=2000 in Figures 5.22 and 5.23. Here, the grid distribution is 101x 11 which is considerably lower than that of references. The results of Morihara [75] and AbduWou [87] have been induded for comparison. AbddNour solves the stream fimction-vorticity form of the Navier-S t okes equations with implementing a second-order boundary conditions. Although this distribution is not consistent with the results of the previous investigators it has been illustrated here to show the very recent attempts in this regard. The distribution shows abrupt jump at the ialet of the channel which is due to abrnpt progress of the centerline velocity as soon as it enters the channel. The renults of Carvalho et al [88] have also been shown here. Their method of solution, an integral transform method, is applicable to high Reynolds numbers, Re + W. Moreover, the solutions in the entrance region approach asymptotic values when Re -+ W. As is meen, the results of the present solution at high Reynolds numbers show excellent agreement with the results of the limiting values.

Finally, we examine the performance of the analogy with respect to convergence. In this regard, the entrance cases of Chen and Pletchex (451 who solve for all speed flows are selected and th& reported results are compared wit h the results of our analogy. The test cases include four cases with inlet Mach numbers of 0.05. Grid distributions of 21 x 11, 21 x 11, 31 x 11, 41 x 11 are used mth nondimensional channel lengths of 2, 4, 30, and 3000 to solve for Reynolds number of 0.5, 10, 75, and 7500, respectively. Here, the Reynolds number is based on the inlet vdouty and half width of channel. The mesh distribution is a non-dom one which is not similar to those used by the references. Table 5.5 presents the results of this comparison. The number of thesteps to achieve the convergence criterion of IO-' for aii dependent variables is significantly lower for the analogy-based procedure. CHAPTER 5. TWO-DIMENSIONAL RESULTS

U distribution on œnlrline grid

l1B

Figure 5.22: Centerline velocity distribution for incompressible and compressible flows in entrance region, Re=2000.

Convergence History of F to' I I . Inle( Madl No.

- M==.10 1 -.-.- k0.05 - M-4-01 ...... ~.001 1 + + lnaamproooble

---..

Figure 5.23: Comparing the convergence histories for entrance flow , Re=2000. However for Re=0.5, the number of tirne steps is not as low as the others. In this regard, attention must be drawn tond the selected grid. This grid does not appropriately reveal the flow pattern in such a short 1ength. Rapid convergence is recovered at higher Reynolds numbers.

-7 Reynolds 0.5 10 75 7500 Grid 21x11 21x11 31x11 41x11 Chen and Pletcher [45] 25 42 87 200 This work 6 3 3 3

Table 5.5: The number of time steps to achieve the aiterion in compressible en- trame flow, M=0.05

5.3.3 Converging-Diverging Nozzle Flow

The third test problem is hyperbolic planar converging-diverging nozzle flow. The geometry of this nozzle is seen in Figure 5.24. This symmetric planar nozzle ha9 an aspect ratio of AR = =2.0. Compntations were performed for this mode1 using the Euler equations. In this test, 7=1.4 and T = 75°C were considered for the 0uid. The results have been obtained for the upper half-nozzle using a 51*11 domgrid distribution. The grid lines are depicted by dotted lines in Figure 5.24. Slip boundary conditions were applied at the walls. Back pressure and mass flow were speded downstream and upstream of the nozzle respectively. Temperature was also spedied at the inlet and the energy equation was dosed for boundary control volumes at the outlet with zero difhisive flux. Adiabatic boundary conditions were applied to the walls. There was no condition on the y-component CHAPTER 5. TWO-DIMENSIONAL RESULTS

Figure 5.24: Hyperbolic convérging-divérging nozzle coconfigoration, AR=2.0. of mass flux at either inlet or ait. In this study the mass flux and density variables are normally nondimensionalized with the values of the parameters at the inlet of the nozzle when they are plotted in figures. Pressure is nondimensionalized with the badc pressure, Phck. For the initial condition, the flow is considered to be at rest and having ambient pressure at aJl grid locations.

This test problem is solved for incompressible flow and for a number of low Mach number compressible flows. The details of the procedure is quite simiiar to those for the cavity problem. The low Mach numbers are M=0.001, 0.01, 0.05, and 0.1 which approximate incompressible flow. Figure 5.25 shows the density distribution along the centerline of the nozzle for these test cases. As seen at lower Mach numbers, the density becomes uniform. Figure 5.26 compares the distribution of other flow parameters along the nozzle centerline for the tested low Mach nnmba flows. These sub-figures iliustrate the consistency.of the other field parameters with the density field. Finally, the convergence histories for Werent low inlet Mach numbers have been depicted in Figure 5.27. It shows that the trend of convergence is similar to the trend in the cavity problem with slight différences. Since the pressure field changes are much sensitive at higher Mach numbers, the dect indirectly shows up in the convergence history. Although at the very low Mach number of M=0.001 the result is definitely identical with the incompressible CHAPTER 5. TWO-DZMENSIONAL RESULTS

on -mRO distribution centerline grid

Figure 5.25: Density distributions for four different low inlet Mach numbers.

O=?% -5 O 5 ;O x position x position

Figure 5.26: Mach distributions for four dinerent low inlet Mach numben. CHAPTER 5. TWO-DIMENSIONAL RESULTS 166

one, a depatture is seen for highet Mach number flow where the compressibility is slighkly effective. This is more serious for M=0.1. Sesterhem et d [44] has examined similar cases of Iow Mach namber flows in a quasi one-dimensionai Lavai nozzle using 100 equally spaced control volumes. A total of 22 time steps ras needed to achieve the speded criterion by changing the CFL numba hm100 to 2000. Their method was restricted to only a one-dimensional study.

Figue 5.27: Comparison of convergence histories for low inlet Mach number flows in nozzle.

5.4 Compressible Flows

Contrary to the previous sections, highly compressible flows with considerable changes in density are investigated in this section. Highly compressible flows are mainly divided into subsonic, transonic, sapersonic, and hypersonic flows. There CHAPTER 5. TWO-DIMENSIONAL RESULTS 167 are no shock waves in subsonic regimes while shock discontinuities become a major problem in treating the other regimes of flow. We start our testing with subsonic flow regimes. Lata, this examination is extended to inchde the other flow regimes. In consequence of onr studying compressible cavity and nozde flows, these are two which are selected fiom the previous sections to be examined once more for high speed compressible flows. Dinkrent CFL nambers have been used to solve the problems in this section. They are provided for each case.

5.4.1 Compressible Cavity Problem

To demonstrate the performance of the present method in solving compressible flows, the cavity problem is recded. This mode1 problem was introduced in Section 5.2.1 and it has already been solved for both absolute incompressible flow, Section 5.2.1, and pseud~compressibleflow, Section 5.3.1. The cavity problem is once more tested here to emphasize the performance of the compressible algorithm of the code in solving high Reynolds number flows with high subsonic speeds. Section 5.3.1 and Ref. [55] provide more det ails.

In order to increase the compressibility dects within the cavity, the velocity of the lid is gradually increased. The efFect is to raise the Mach number. Many cases were examined at different Mach numbers up to sonic speeds. Here we select M=0.8 to represent the related results. This test problem was also studied for severd Reynolds numbers including high Reynolds number of 3200, 5000, and 7500 with a grid distribution of 51x 51, 51x51, and 71x 71, respectively. Figures 5.28 to 5.30 illustrate the centerline velocity distributions for these selected problems. The oenterline velocities have been nondimensionalized by the velocity of the moving lid in ail cases. The resdts have been compared with the incompressible CHAPTER 5. TWO-DIMENSIONAL RESULTS

Compressibility study in cavity (Re-3200) U distribution on the vertical center line

u velocity V distribution on the horizontal center line 0.6 1 O Ghia129x129(72] 1

O 0.5 1 x position of nodes

Figure 5.28: Compressible cavity, Re=3200, grid 51x 51, Md.8. Compressibility study in cavity (Re=5000) U distribution on the vertical center line

u velocity V distribution on the horizontal center line 0.6 1 O Ghia257x257 [72] 1

O 0.5 1 x position of nodes

Figure 5.29: Compressible cavity, Re=5000, grid 51x 51, Me0.8. Compressibility study in cavity (Re=7500) U distribution on the vertical center line

u velocity V distribution on the horizontal center line 0.6 1 O Ghia257x257 V2J 1

O 0.5 1 x position of nodes

Figure 5.30: Compressible cavity, Red500, grid 71x71, MeO.8. CHAPTER 5. TWO-DIMENSIONAL RESULTS 171 results of Ghia et al [72]. At the same the, they have been compared with the incompressible results of the present method. There was a discussion about this agreement in Section 5.3.1 which is not repeated here.

These excellent results represent the performance of the method in solving Navier-Stokes equations for compressible flow with implementing a closed boundary condition. In the following sections, this performance is tested for solving Euler flow equations with 0th- form of flow boundary conditions.

5.4.2 Converging-Diverging Nozzle Problem

In this part, converging-diverging nozzle configuration is tested for several different types of moderate and high compressible flows. They indude subsonic, supersonic, and their combinations. Since most of the results of the current method in this section is compared with the analytical solution of the one-dimensional flow, the nozzle configuration has been selected in such a shape that its results are closer to the one-dimensional flow.

Subsonic Flow

The fùst step in subsonic flow investigation is to study the dect of nozzle oonfigu- ration in the accuracy of the solution respect to the ondimensional exact solution. This study is directed tkough an isothermal flow condition. The nozzle configuration in Figure 5.24 with ail side conditions presented in Section 5.3.3 is recded. Here, the flow is isothermal which eliminates the &ect of temperature field changes. Figure 5.3 1illustrates the distribution of Mach and pressure on the centerline of the nozzle. The highest subcritical Mach nnmber in isothermal flow is las than that in isentropic flow. In an isothermal region, the flow becornes critical if M > (?)-If2. CHAPTER 5. TWO-DIMENSIONAL RESULTS 172

For air, this limit is M > 0.85. Tbhas been emphasized in Figure 5.31 by drawing a horizontal ddline in the Mach plot. There is a good agreement between the results of the current method and that of the one-dimensional solution.

Convergent-Divergent Noule Resuits Mach

Figure 5.31: Mach and pressure distributions for an isothermal nozzle, Min a0.25.

Next, we change the nozzle configuration in order to study the &ect of tw* dimensionality of the configuration and flou. In this regard, the nozzle configuration is changed to a more realistic shape which is fm fiom one-dimensional asswnption. The geometry of this nozzle is seen in Figure 5.32. This symmetric planat nozzle bas an aspect ratio of AR = =2.035. We present the results of solving the Euler equations for this test problem and compare them with the available one-dimensional exact solution [89]. CHAPTER 5. TWO-DIMENSIONAL RESULTS 173

The test case was examined using a 31x11 dorm grid distribution. The grid lines are depicted by dotted lines in Figare 5.32. This figure abillustrates the Mach contours within the flow field. Since the flow fieid is totally snbsonic, these contours are convex in the convergent part and concave in the divergent part. These schematic patterns are in good agreement with those of other predictions, Oswatitsch and Rothstein [go].

The isothermal Mach number and pressure distributions on the walls and the centerline are shown in Figure 5.33. These results are compared with the exact se lution of the one-dimensional isot hermd flow t kough a nozzle [89]. There is a good agreement between the two solutions. However, the &ect of the twedimensionality of the flow is much more critical in this test case t han the one presented in Figure 5.31. The geometry of the nozzle shows how the one-dimensional solution could be far from the real solution. A cornparison between the convergent and divergent parts shows that the deviation of the numerical solution fiom the one-dimensional solution is dinerent in these two regions. The convergent part is much farther fiom the one-dimensional solution than the divergent part. This would cause more deviation fiom the one-dimensional solution in the convergent part which is consistent mth the obtained results. This model was tested for longer lengths while retaining the aspect ratio constant. The results showed that the distribution dong both walls and dong the centerline approached the one-dimensional solution.

After this preliminmy study on isothermal flow and the importance of the nozzle geometry, we recd the nozzle in Figure 5.24 to test the compressible algorithm foi subsonic non-isothermal flows. In this regard, the inlet velocity is appropriately ked upstream to capture three high subsonic Mach numbers of 0.5, 0.8, and 0.95 at the throat. The density, Mach, and temperature fields are depicted in Figures 5.34 to 5.36. These figures present the distribution at the walls aad dong the CHAPTER 5. TWO-DIMENSIONAL RESULTS

nodal M contour Iiin Ibw field

Figure 5.32: Mach contour lines in a converging-diverging nozzle with AR=2.035.

Convergent-Divergent Nozzle Results lsothennal Mach

1 1 1 t 1 I 1

0.8 - 00 1Drduliari ,.. ,.. uppwd œ"~Olid -kvverwdl

0.2 I 1 1 1 I 1 O 5 1O 15 20 25 30 35

Pressure 1.1 I I I l I I

1 -

g3 0.9 - z 0.8 -

0.7 -' L 1 1 I I I O S 1O 15 20 25 30 35 nodes in X-direaion

Figure 5.33: Isot hermal Mach and pressure distributions for the nozzle presented in Figure 5.32 with Mh =0.3. CHAPTER 5. TWO-DIMENSIONAL RESULTS 175 centerüne of the nozzle. These resdts have been compared with the exact solution of the one-dimenaional isentropic flow approximation tkough a nozzle. There is an excellent agreement between the two solutions. At first, it might be expected to see the onedimensional solution alaays between the centerhe and rd solutions. This is heuntil the throat reaches sonic speed. Then, the throat Mach number becornes highly sensitive to siight changes in inlet Mach number. Cornparhg the inlet Mach numbers for the cases with throat Mach numbers of M=0.8 and 0.95 revealr the high sensitivity of the throat values near sonic speeds. In another words, numerical mors will cause significant deviation fiom the exact solution near sonic speeds. On the other hand, a look at the geometry of the nozzle reveais how far it is fÎom being one-dimensional. The two-dimensionality of the problem is another reason which causes deviation fiom the ondimensional solution.

In addition to the two factors of the nozzle geometry and the sensitivity at high Mach numbers, there is the mesh size factor which aEects the accuracy of the numerical solution. The dect of this factor ha9 been ilîustrated in Figures 5.37 and 5.38 for the nozzle problem with MthtOote0.95. This nozzle problem has been tested for thdifferent mesh sizes of 101x11, 51x6, and 25x4. The Mach distributions on the centerline have been compared with the one-dimensional solution. As is seen, her grid shows better accuracy comparing with the one-dimensional solution. The Mach contour lines have been depicted for these three different mesh sizes in Figure 5.37. Finer grid distribution demonstrates smoother distribution around the throat comparing with coarser distributions.

Generdy speaking, Figures 5.34 to 5.36 present excellent pafonnance of the compressible aigorithm for solving high subsonic compressible flows. It is essential that the obtained solution presents a symmetric distribution left and right of the throat section. This symmetry is well demonsttated within these plots. Identical CHAPTER 5. TWO-DIMENSIONAL RESULTS

RO distribution on center grid and walls 1.05 1

Figure 5.34: Density distribution for three different throat Mach numbers.

MACH distribution on center grid and walls

œnîaIhe

nodes in X-direction

Figure 5.35: Mach distribution for tkee different throat Mach numbers. CHAPTER 5. TWO-DIMENSIONAL RESULTS

T distribution on center grid and Wls

'.Oe@ '.Oe@

Figure 5.36: Temperature distribution for three different throat Mach numbers. values of the parameters at the inlet and out!et of the domain supports the ability of the code in solving the Euler flow equations.

For the boundary condition implementation, the pressure was specified downstrearn, which is why all three lines of pressure distribution are matched at the exit of the nozzle. The upstream pressure was computed by the code. k an Euler flow, both upstream and downstrearn should attain the same pressure if th& aspect ratios are the same. This was a good test for checkhg the pressure drop in this method. The pressure drops which are presented on Figures 5.31 and 5.33 report the pressure loss from inlet to outlet. These low numbers are another indication of the accuracy of the method for this subsonic flow. CHAPTER 5. TWO-DIMENSIONAL RESULTS

Figure 5.37: The &ect of mesh size on the Mach contour distribution.

Figure 5.38: Mesh size &ect on the accnracy of the numerical solution. Wied Subsonic-Supersonic Flow

In this part, the shock capturing capability of the curent method is investigated by testing mixed subsonic-supersonic flow t hrough a converghg-diverging nozzle. There are two limiting cases for convergent-divergent nozzle flow when the Mach numba of the throat is unity. The flow in divergent part of the nozzle can either be Mysubsonic, with symmetric distribution of flow variables respect to the throat section, or supersonic, with a smooth deaease or increase of the flow parameters from inlet to exit. Between these two limiting cases, the flow is not stable unless there is a shock in the divergent part of the nozzle. In order to generate a shock in the nozzle, it is necessary to have sonic speed at the throat. In this regard, the idet stagnation pressure is increased nrtil the flow is choked. Then, the back pressure is decreased to a lower value than the inlet. This results in s normal shock wave within the divergent part of the nozzle. The ratio of & determines the location and the strength of the shock.

The nozzle figure is as before in Figure 5.24, while the grid distribution has been changed to a dorm 51x9 distribution for the whole nozzle. Initidy, the inlet stagnation pressure was specified at the inlet and the exit pressure was selected in such a mannes that a pressure ratio of &=0.7932 produced shock at the particular section *=1.31 with strength, *=2.575.M. t The results of the current method are depicted in Figures 5.39 and 5.40 and compared with that of the one-dimensional exact solution and Karki (911. Karki solves the test case using a quasisne-dimensional algorithm. Schneider and Karimian [12]have also solved this problem using a quasi-one-dimensional algorithm without mentioning the ptoiile of the nozzle. Since the continuity equation is sacfificed to speufy mass at the inlet, the control volumes which are placed at the inlet do not necessarily conserve mass. C'TER5. TWO-DlMENSIONAL RESULTS

MACH disttibutbn for supersorrk nozzle, isentropic 14 i Redts 14 - -eiritor lin *nlr 14 00 em1Ordution 12- ++ KYki[Ql]

QI-

Figure 5.39: Mach distribution in nozzle with M-x1.67.

R 10' P distribution for supersonic noule, isentropk k

Figure 5.40: Pressure distribution in nozzle with M- =l.67. As obmed in Fi- 5.39, this defèct causes some discrepancies in the dution doae to the inlet. The exact solution har been computd right at the noda and these values are shown by circies. The exact location of the shock ia not necessady at nodes but somewhae between tao neighboring nodes in the vicinity of the shock discontinuity.

As seen, the shock has been capturecl very well in the divergent part despite using a coarse grid. The shock position and the exit Mach number have been pre àicted very wdcomparing with the one-dimensional solution. Lien and Leschziner [53] have solved a similar inviscid nozzle flow using either the quasi-one-dimensional or twdimensional models in their aU speed flow solva. They show that the ta+ dimensional results are farther than the quasi-one-dimensional dution to the onedimension$ exact solution around the shock. consistent with th& eqerience, oiir resdts ahshow that the shock has been slightly smeared in the front of the shock and a minor undershoot is obsented behind it. The source of this oscillation is fiom the temperature integration point caldation which is appmxhated by a bilinear interpolation in this case. The test problem was solved with a msirimum Courant number of 1.25. This maximum occurs at the left-hand-side of the shock. A totd of 222 time steps wae executed to achieve the RMS of IO-' for most of the dependent variables. It should be noted that no special treatment has ben speafied for the distribution in the vicinity of the shock wave.

In the second test, this nozzle configuration is ernmined for floa with stronger normal shock wave, Figures 5.41 to 5.43. In addition to th&, the oscillation around the shock wave is investigated. In order to generate a stronger shock, the exit pressure is redud to &=0.57. This produces a shock at section Sd.837, with a strength of *=3.74. The maximum Mach number fa thk case is Mw=2.1 which is only 0.1 less than the mnlcimum Mach number for a funy supersonic flow Please Note

UMI CKAPTER 5. WO-DIMENSIONAL RESULTS

MACH disaibution on lower and upper wails

Figure 5.50: Mach distributions on the walls of channel with and without using convecting momentum equation, Mi,=0.5.

Tkansonic Flow

In this part, the method is examined for more highly compressible flows, transonic flow. The test configuration and its grid arrangement are as before while the inlet Mach number of the flow is increased to a supercritical transonic case, i.e. Mim=0.675. This Mach nnmber causes s supersonic region in the solution domain which is terminateci by a shock. Figure 5.51 depicts Mach contour lines within the domain. These lines are no longer symmetric respect to the mid-chord line because there is a shock on the bump. The Mach contour Iines are also not perpendicalar to the wall donnistream of the bump, as they are upstream, because the flow becornes rotationai upon passing the shock. Figure 5.52 demonstrates the distributions of Mach numba on the uppa and Iowa walls of the channe1 and compares these nodal M contour lines in flow field

Figure 5.51: Mach contours for transonk flow in a Channel with a bump, Mh=0.675.

0.3 1 1 1 b 1 1 I O 0.5 1 1.S 2 25 3 X-di rection

Figure 5.52: Mach distribution on the walls of ehannel and eomparing with [92] using 89 x 33, [33] using 60 x20, and [32] using 67 x 22 non-dorm grids, Mh=0.675. CHAPTER 5. !ïWO-DIlMENSIONALRESULTS 191 with the resuits of other workers. The agreement of this work with the resdts of Eidelman [92] is very good although a little disparity is observed behind the shocl.

In orda to have a quantitative cornparison as wdas qualitative one, the location and magnitude of the captured shock have been determined and represented in Table 5.6. The results of the cment work are compareci with those of [33,92,93]. Ni [93] developed a mdtigrid scheme in the context of the density-based compressible Bon algorithm. His rdtsare more accurate. Since the shock for this work has a bit of spread its location has been calculated based on the midpoint between the upstream and downstream of the shock. The resdts of the current work compare favorably with those of the other researcher.

.. Present method Ni (931 Eidelman [92] Karimian [33] Bnmp Chord% 75% 72% 72% 75% Max. Mach No. 1.29 1.37 1.29 1.25

- - - -- Table 5.6: Location of the shock in percentage of the bump chord length

The initial condition was F=G=O for this case. The total of 50 time steps was required to reach the convergence criterion, i.e. RMSCIO-~for aii dependent variables, with A8=0.1.

Supersonic Flow

The finai case examined for the bump Channel flow is for supersonic flow. The configuration of the test model remains the same except the bump thickness is reduced to 4%. The grid mangement for this case b different nom the previous one. It is a non-dom 77x21 grid distribution which is shown in Figure 5.53. A coarser distribution has been considered for upstream of the bomp because the flow in that region is not affkcted by the bump. The grid distribution dong the dianne1 over bump and its downstream are reiatively dorm, however, non-dom grid distribution has been considered for the cross sections to have bettes resolution of the shodc on walls. The inlet Mach nnmber of Mk=1.65 is selected to be compatible with the results of other workers.

Grid distribution for supersonic flow

Figure 5.53: Grid distribution for supersonic flow in a channei with bump.

The isobar iiaes in Figure 5.54 show the position of the shock waves inside the channel. As observed, two oblique shock waves are formed at the corners of the bump. The 1eading edge shock is followed by expansion waves reflected from the bump's body. This shodt later strikes the upper wall and is reflected back by the wall into the expanding flow field. The trailing edge shock leaves the computational domain âom the exit boundary after intersecting the expansion waves. The numba of intersections and reflections indicates that this supersonic problem is a difficult one.

The Mach distributions on the lower and uppa walls are seen in Figure 5.55. The results of the curent method are compared with those of the previous rderences for which the grid arrangements were different fiom the previous test cases. As seen nodai P contour lines in flw fieid

Figure 5.54: Mach contours for supersonic flow in a Channel with a bump, Min=l-65.

MACH distribution on lower and upper walls

-present method

'" ' - EWman 6 Coldia (921

1.8 - O O O Kahian & Schneider [SI x x x Karki 6 Patankar (321 1.7 -

1.5 -

1.4 - lawer wd 1.3 -

1.2 I I m I I O O .5 1 1.5 2 25 3 X-di rection

Figure 5.55: Mach distribution on the walls of channel and comparing with [92] using 89 x 33, [33] using 60 ~20,and [32] asing 67 x 22 non-dona @ds, Min=l.65. fiom the figures, the agreement of the results on the lower wall h better than that of the upper wail, however, for the lower wd, the flow near the channel exit displays some osdatory behaviour. This behaviour identifies the need for improvement in this region of the flow. Although the cause of this is not entirely ciear, the intersection of the reflected shock with that corner might be an important factor.

The problem was started with initial conditions of Fo=500 kg/(m/s2), Go=O.O kg/(m/s2), P0=86,100 Pa, and To=300 "K.This non-zero Fo speeds up the convergence. The number of time steps to reach the convergence aiterion for RMSCIO-~ was 220. The maximum Courant number used for this test case was 0.86.

5.4.4 Supersonic Flow in a Channel with a Ramp

The purpose of this section is to study the performance of the cment method in solving flow with strong oblique shock waves. The test model is a channel with a ramp mounted on its lower wail. This problem is very appropriate because of the availability of the exact solution. This enables us to evaiuate the accuracy of the method. The geometry and grid distribution for this problem are shown in Figure 5.56. The longitudinal and transverse lengths are 1.3 and 1.2m, respectively. The ramp angle is 21.57-deg and its leading-edge is located at z=0.3m. The ramp angle creates an oblique shock wave with 45deg angle if the inlet Mach number is Min=2.5. This shock has an strength of M2/Ml=0.628, P2/Pi=3.497, and T2/Ti=1.508. This shock will leave the solution domain without colliding the solid waUs. A non-uniform grid dis tnbution of 41 x 37 has been used for the computation, Figure 5.56.

Figure 5.57 depicts the Mach contour lines in the solution domain. As is seen, the main changes of the flow parameters occurs across the shock wave. The oblique CWTER5. TWO-DIMENSIONAL RESULTS

Figure 5.56: Grid distribution within a Channel with a ramp.

nodal M contour lines in fbw field I

Figure 5.57: Mach contour lines plot for ramp with Mim=2.5. CHAPTER 5. TWO-DIMENSIONAL RESULTS 196 shock wave leaves the domain with an angle of 45-deg which is identical with the analytical solution.

The details of the solution inside the channel have been psovided in Figure 5.58. This figare shows the Mach distribution at a constant height of y=0.15m inside the channel and compares the results of the carrent method with that of Karimian and Schneider [33]. The cornparison of these results with the exact solution shows that the shock wave has been captured within almost 6 nodes by the bath methods. However, the current method shows an overshoot behind the shock. This is not seen in the results of the reference. This might be the dect of damping mechanism which is used by the reference. The results demonstrate the ability of the current method to capture strong oblique shock wave without using any irnplicit artificial viscosity or damping mechanism in its algorithm.

MACH distribution at ~4.75

Exact Solution present method 41x37 x x x Kanmian & Schneider 45x30 (33)

Figure 5.58: Comparing Mach distribution resdt at height y=0.75m with 0th- results in a duct with a ramp. CIEAPTER 5. TWO-DIMENSIONAL RESULTS 197

In this test problem, the selected convergence criterion of RMS<10-5 was de creased to RMS

The problem was started with initial conditions of Fo=500 kg/(m/s2), Go=O.O kg/(m/s2), Po=86,100 Pa, and To=300 OK. A non-zero Fo speeds up the convergence. The problem was solved for an inlet Courant number of about 0.55. The total number of 416 the steps were needed to satisty the RMS<1W6 for F and G. This number is reduced to 265 if RMS<10-'.

5.5 Cornparison Between Velocity and Momen- t um Formulations

In this section, we present a cornparison between the momentum-based and velocity- based formulations for high speed compressible flow. It is noted that there is no significant difference between the velocitjj-based and momentum-based formulation for absolute incompressible flow where density is a constant. The discussion on the paformance of the corrent method in solving pseud

A fiterature review of the methods which solve for incompressible and compressible flows at all speeds was presented in Section 1.2.4. Zienkiewicz and Wu [49] solve subsonic and supersonic flows for a number of applications using an expliut or mmi-explicit finite-element method. They solve the non-consetvative fcaa or" CHAPTER 5. ZWO-DIMENSIONAL RESULTS 198 the goveming equations and admit that this may lead to dXerent shock behaviour than that involving the conservation form. They do not report the convergence histories for their steady-state solutions. Van Doormal et al [50] ülustrates the appiicability of segregated methods for solving flow at dl speeds and evaluates the relative convergence behaviour of these methods for solving the ondimensional and two-dimensional lamina compressible flows. This provides a good potential to compare the pedormance of the current method with that of the segregated ones. However, this cornparison has been done for twdimensiond studies with a different segregated approaches. Karki and PatanLat [32] have developed SIMPLE- based methods to solve for flow at all speeds. The density at the integration point is always upwind-biased. This provides an artificial damping which allows for the successfid computation of transonic and supersonic flows. Their resnlts have ben compared with the results of the current formulation as illustrated in Figures 5.39 to 5.42, for the converging-diverging nozzle problem mth shock, and Figures 5.49, 5.52, and 5.55, for flow in a channe1 with a bump. Their results show smearing in the vicinity of shodrs due to excessive numerical dissipation. They do not report the convergence histories of th& solutions.

Chen and Pletcher [38, 451 employ explicit/implicit second-order and explicit fourth-order smoo t hing terms in t heir time-marching met hod to solve transient flow and flow at all speeds. Their work is fully impliut and d variables, (u,v,p,t), are computed simultaneously. They report the convergence behaviour of th& solutions which we compared with that of the current work. This comparison has been presented in Table 5.4 for solving the compressible cavity flow and in Table 5.5 for solving subsonic entrance flows. GeneraUy spealing, the momentum-based procedure provides fater convergence and higher stability.

In addition to the above methods which solve for flow at ail speeàs, it is the method of Karimian and Schneider [27, 331 which enables us to provide a direct comparison between the performance of the momentnm-based formulation and that of the vdocity-based formulation. As ras mentioned in Section 3.6, a ditndty with their method is the requirement of an explicit damping mechanism. This does not permit a neat and dear cornparison between the two formulations. In a one- dimensional investigation, a cornparison was performed by direct modeling of their method without considering the damping mechaniom, Section 3.6. For the tw* dimensional investigation, there remains a question of whether the velocity-based formulation can retain its characteristics of cmvergence and stability if the darnping mechanism is elirninated from the scheme. Karimian [94] states that th& velocity- based procedure would diverge without inelosion of the damping mechanism. This statement results in several important concIusions which are discussed subsequently in this section.

Table 5.7 provides an informative comparison between the velocity-based and momentum-based formulations for highly compressible flow problems which indude flow through channels with a bump and with a ramp on the lowa boundary. The nnmber of iterations, NOI, and tirne step, A@, are tabulated based on an RMS convergence criterion of IOm5for the momentum formulation and of IO-' for the velocity formulation. Alt hough the initiai and boundary conditions are the same for both formulations, the grid resolutions and distributions are not exactly identical.

Despite different RMS criterions for velocity-based and moment um-based formulations, a numba of points codd be resulted if we assume that they are the highest criterions to which steady-state solutions are botained in each of these formulations. For the subsonic flow, it is the moment- procedure which shows faster convergence in comparison with the velocity procedure in meeting a definecl RMS. For transonic and supersonic flows, this fat convergence is diminishes with inmeas- momentum-based formulation velocity-based formulation II withnodamping 1 with artificial damping this research Katimian and Schneider [33] R.M.S 0.00001 0.001 Subsonic Bump, Mi,=0.5 NO1 8 8 AtY huge 1.0 Bansonic Bump, Min=O .675

II ~&rsonic Bump, M, =1.65

Table 5.7: Comparative study of the convergence histories between the velocity- based and momentum-based formulations. ing Mach number. For supersonic flow, the performance of the vdocity procedure with damping is superior to the momentun procedure without damping. Although both procedures start with smd the steps, the velocity-based approach multi- plies it &es a number of preliminary steps. The time step inaease speeds up the convergence toward the steady-st ate solution. If the time step is inaeased for the CHAPTER 5. TWû-DLMENSIONAL RESULTS momentam-based formulation which is fiee from any explicit damping mechanisma, it mothandle the acceleration produeed by the force of the initial flow and of the boundary condition implementation. Generally speaking, the moment am-based formulation has several advantages with respect to the velocity-based formuiation without damping. The use of the momentnm-component formulation does increase the stability of the method in cornparison with the veloeity-component formulation which requires a damping mechanism for all time steps in order to converge. Some under/overshoots are observed around strong shocks in the momentum-based formulation but do not cause serious diEculties in the convergence if the time step is smaU enough. It is the lack of a damping mechanism which does limit the time step to smd values. Indeed, the momentum-based formulation does need improvement if it is desired to achieve faster convergence. This provides one possible area for future research.

At the end, it is noted that the advantages of the momentnm-based formulation in obtaining better convergence without using a damping mechanism are supported by a number of factors. Two important factors were discussed in the research motivation. They are the stability of the mass flux in passing through shocks, Section 2.4.2, and the reduction in the linearization requlements, Section 2.4.3. However, the importance of each of these factors in promoting cannot be fUy evaluated separately.

5.6 Closure

The proposed momentum variable procedure was examined for many different tw* dimensional flows including incompressible, pseudo-compressible, and subsonic to supersonic compressible flows. The method showed excellent performance in solving CHAPTER 5. TWO-DIMENSIONAL RESULTS 202 flow at ail speeds. There was no CFL number limit in solving incompressible and subsonic flows. The solution converged very rapidly aithin a small number of iterations. However, the convergence slowed doan in supersonk flows with CFL numbers las than one. Despite not employing any explicit artüicial viscosity or dissipation and in spite of using marse grids, excellent solutions wae obtained in cornparison with the work of other refêrences. However, the formulation stU needs improvement to increase its convergence in cornparison 4th the velwty- based formulation where damping is employed. The proposed formulation showed excellent performance in solving very low Mach number flows with respect to both the solution accuracy and the convergence history. Chapter 6

Concluding Remarks

6.1 Summary

The difference in the nature of compressible and incompressible flows has resulted in the development of numerous numerical techniques to deal with each of these two types of flow separately. There have alPo been sorts to develop algorithms capable of solving both compressible and incompressible flows. In this thesis, a new tw~dimensionalunsteady viscous computational algonthm has been developed to solve flow at all speeds. A strong motivation for this development has been to explore the use of momentum component vaxiables instead of the velocity components usually used in all speed solvers. Several reasons were behind this. A significant one of which is the strong analogy that exists between the two kinds of flow when such variables are used. This analogy, developed in this thesis, pdtsincom- pressible flow methods to be applied to compressible flow problems. In addition, using the momentum components improves stability amund shocks and reduces the linearization difficulties. The two-dimensional Navier-Stokes equations were selected to examine the performance of the proposed method. The method was developed using a control- volumebased nnitdement scheme. The finite-element part of this scheme pre vides the benefit s of finite-element geome tric flexibility while the advant ages of conservative discretization procedures are provided by the use of the eontrol-volume formulation.

Initidy, this proposed new direction was explored for one-dimensional flow modeling. In this one-dimensional investigation, momentum, pressure, and temperature were selected as the dependent variables in a colocated grid arrangement. The governing equations were treated in conservative manner. The integration point equation for the momentum component was daived by approximating the non-conservative form of the momentum equation at the integration point. Rear- rangement of t his equation enabled the integration-point momentum, or convected momentum, to be determined. If only this convected momentum component was employed in the conservative form of the discretized equations, it was found that a special form of the pressure checkerboard problem resulted. A number of difE'erent treatments to overcome this difficulty were examined. In this regard, a new integration point equation was derived by the combination of the momentum and continuity equation mors. This new equation was named the convecting equation which represented the convecting momentum component. The influence of the continuity error was examined by applying an appropriate factor. The use of bot h the convected and convecting momentum equations removed the possibility of the pressure-velocity decoupling problem. The resulting one-dimensional algorithm was then validated for incompressible flow by using several source/sink test cases. The algorithm was &O validated for compressible %on by comparing with the analytical results for the shock tube problem. Finally, the performance of this one-dimensional momentum-component formulation was compared with that of the velocity-component formulation by direct modeling of the latta formulation.

Next, the procedure was extended for solving tw*dimennional flows. Convected and convecting integration point equations were again derived by approximating the non-conservative form of momentum and the consenative form of continuity equations at the integration points. The stream-rrise treatment of the convection terms had an important impact in achieving the correct physical modeling of the flow. The tw~dimensionalalgorithm was validated for many dinerent flows induding incompressible and compressible ones. The flow models were dassified into the three categories of incompressible, pseud~compressible,and compressible flows. The results of the developed method were compared with those of several velocity-based procedures which solve flow for all speeds. The good to excellent performance of the method in achieving reliable results on coarse grids without using the explicit artüicial viscosity and damping mechanisms are noteworthy. The folowing sections present the main conclusions of this work and tecornmendations for future research.

6.2 Conclusions

The main purpose of this PhD research has been to explore and hdthe advantages and disadvantages of the momentum-variable procedm in cornparison with the velocity-based procedure. The potential advantages of the momentum-based formulation motivated this research: the existence of a ffow analogy between the governing equations for compressible and incompressible flow, the stability of the mass flux throagh shocks, and reduced difficuity in linearjzing the nonlinear terms.

It is dmost impossible to recognize the degree to which each potentiai advantages ie involved in gsiniag the superior malta for the rnomentum-bd procedure. In a one-dimensional unsteady flow investigation, it was shown that the stability of the mass flux passing through shocks was not a crucial fmtor. However, this assessrnent was much more diffxcult in the twdimensiond study. Considering this difliculty, the advantages and disadmtages of the momentum-variable procedure are discussed in the following paragraphs mainly without reference to the factors which cause such advantage or disadvantage.

The conclusions are divided into two parts. In the hst part, the characteristics and capabilities of the new method are presented. This part is developed mainly by comparing with exact solutions and the results of other workers. The second part compares the advantages and disadvant ages of the moment um-variable procedure with those of the velocity-variable procedures. This part is conduded by a direct cornparison with the results of the velocity-based procedures which solve flow for aU speeds. Based on the method development and the demonstrated results of its application, the fust part of the conclusions are

1. The benefits of the finite-element bais of the method facilitates domain discretization, and no difficulty was encountered in fitting the grid to the domain.

2. The integration point equations which were derived provide correct physical behavior of the flow variables. This ability was shown not only by the numer- idsolution but also by an andytical investigation. The Peclet and Courant numbers are two parameters which provide flexibility of the formulation for a wide range of flow parameters.

3. The proper consideration of the role of velocities at control-volume surfaces resulted in two sets of integration point equations; convected and convecting ones. The correct application of these equations generated a strong connection between the velocity and pressure fields and removed the possibility of the cheekerboard problem. This capabity was demonstrated by nnmerous tests in both one-dimensional and twdimensional studies.

4. The method showed good to excelient performance in solving 00w at all speeds including: incompressible and compressible (pseud~compressibleto supersonic), viscous and inviscid, and steady and nasteady. Excellent resdts were obtained for incompressible and subsonic compressible flows despite using coarse grids. The demonstrated results co&m the exdent performance of the developed method in solving low and high Reynolds number flow cases. Very good resulto wae obtained in ~upasonicflows despite the exclusion of explicit artificid viscosity or dissipation functions. This was clearly observed in solving high speed inviscid flows.

5. No CFL number limitation was encountered in solving incompressible, subsonic compressible, and transonic flows. This advantage enabled the present time dependent method to meet the convergence criterion within a rektivdy small number of iterations using large thesteps.

6. The selection of pressure as a dependent variable and developing a pressure- based method enabled the method to be highly robust in treating very low Mach-numba compressible flows. The results of solving the subsonic and transonk flows were noteworthy with regard to the coarse grid distribution and the number of tirne steps required to reach a stdy-state solution.

7. Some difficulties wae encountered in supersonic flow, which generally slowed down the convergence of the solution and limited the CFL number to dose to one. This may be due to the lack of attificial dissipation andior damping functions. Despite these restrictions, the results were fairly good. 8. An andogy between compressible and incompressible iior motivated the use of momentam-components as dependent variables. The pseud~compressible and compressible models were nsed to demonstrate this analogy. The analogy worked wdnot only in deriving similar results for both compressible and incompressible flows but also in providing similar convergence histories.

9. The entire dort the curent work was to develop the method as simple as possible. Complex and time-consuming techniques were avoided so that the conceptual aspects of the method ddbe emphasized. For example, all integration point equations were derived based only on the element nodd

point S. The otbcr t hree int egration point value8 were not involved in deriving the fourth. Despite excluding such a potential benefit, very good results were obtained for highly recirculating flows.

A direct cornparison between the results of the momentum-variable and velocity- variable procedures which solve flow for 811 speeds results in the second part of the condusions. The condusions are categorized based on the characteristics of the different types of flow,

1. There is no significant différence between the velocit y-variable formulation and momentum-variable formulation when density is constant. However, the momentum-based formulation involves and discretizes terms in the momentum equations which are not existent in the momentum equations of the velocity-based formulation. Although t hese terms are mathematicdy zero, they are numerically non-zero. These terms couid be a source of difference between the above procedures. CHAPTER 6. CONCLUDING REMARKS 209

2. The momentum-variable formulation is robust for psendcxompressible 00~s. However, this may be accounted for as a direct advantages of using a pressure- based algorithm. Essentidy, the pressure-based methods which solve compressible flows could be extended to solve for pseudo-compressible flows. How- ever, there is no reason for a pressure-based method to show identical solution and convergence history behaviour for both the absolute incompressible and pseudo-compressible flows. The existence of these capabilities in the results of the momentam-component formulation could be accounted for as a direct dect of the existence of the flow analogy in that formulation.

3. Although bot h the velocit y-based and moment um-based formulation successfully solve the subsonic compressible flows, the latter one generally shows better performance in cornparisou with the former one. This is concluded by comparing the number of iterations which they need to meet the preset convergence crit erion.

4. In supersonic flow, the velocity-based procedure sders from severe oscillations in the vicinity of the shock if no explkit damping mechanism or dissipation is included. This results in major time-step and Courant number restrictions. These oscillations subsequently results in instability and divergence if a lower coavergence aiterion is desired. However, the use of a damp ing mechanism in the velocity-based procedure could result in bet ter st ability and accuracy. The momentum-based procedure, on the other hand, had excellent stability and good accuracy, even without explicit damping, provided the time step and Courant number restrictions were satisfied. 6.3 Recommendations for Future Research

No single method contains all desirable features while avoiding any disadvantages. The following recommendations are snggested for the continuation of fatnre research in this field:

1. There is a need to further investigate the problems encomtered with supersonic flow. This investigation should generally address a number of issues. The ikst is to extend the ability of the supersonic application to solve for higher Courant numbers which will then result in faster convergence. The second is the use of artxcial dissipation techniques and damping mechanisms ro remove undershoots and overshoots around the shock waves and improve convergence . The third is to extend the scheme to more accurate shock capt uring techniques.

2. The possible unification of and the justification for the two convected and convecting momentum component s is ano t her interesting subject for future research. Alt hough mat hematically demonstrated, our unders t anding of t his issue could be firrther enhanced.

3. While the discretization of the domain interior is fdy conservative, the boundaries may not be. The modification of boundary condition treatment toward obtaining a Myconservative solution domain is another subject for future teseuch.

4. A number of other extensions of the present method codd also be parsaed, such as extending the method to: three-dimensional flows, flows with more complicated geometries and boundaries, buoyant flows, heat transfet problems, and other types of finite elements (such as triangles). Bibliography

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3. Raithby, GD.,and Schneider, G.E., "Numerical Solution of Problems in Incompressible Fluid Flow; 'heatment of the Velouty-Pressure Cou- pling" , Numerical Heat 'Ransfer, v.2, pp.417-440, 1979.

4. Patanluu, S.V., UA Calculation Procedure for Tw*Dimensional Elliptic Situations", Numerical Heat 'Ikansfer, v.4, pp.409-425, 1981.

5. Zedan, M., and Schneider, G.E., "A Coupled Strongly Implicit Proce- dure for Velocity and Pressure Computation in Fluid Flow Problems", Numerical Heat Transfer, v.8, pp.537-557, 1985.

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94. Karimian, S.M.H.,UPrivate Communicationsn, Amir Kabir University of Teehnology, Tehran, Iran, September 1996. Appendix A

Element Geometric Relations

In this Appendix, after the introduction, we fist discuss the discretization of solution domain into a number of elements using finiteelement discretization. kt, the details of elemental geometric relations are presented. They provide necessary geometricd transformations between local and global coordinates are derived.

A.1 Introduction

Before discretizing the governing equations, it is necessary to discretize the calcu- ktion domain in some fashion. In the past two decaàes, Werent techniques have been developed for generating computational grids required in the finite-difference or finite element solutions of partial differential equations on arbitrary regions. Sta- bility of the method, convergence speed, and accuracy of the solution are aspects which could be atfected by choosing inappropriate grids. A poorly chosen grid may cause results to be erroneous or may fail to reveal aitical aspects of the true solution. APPENDLX A. ELEMENT GEOMETRIC RELATIONS 224

Generally speaking, the accuracy of finite-difference methods is increased if the underlying mesh fits the region boundaries and is dosely spaced in regions where the solution is rapidly -mrying. 'Ransformation fkom orthogonal eoordinate system to either non-orthogonal or orthogonal boundary-fitted coordinates will ensure the exact boundary fitting and arbitrary grid concentrating; however, this may cause more compIexity in Merential equations which need more numerical works, e.g. Alishahi and Darbandi 1151. Contrary to fuiitedifference method, there is hi tedement method which has been widely selected for diacretiPng the solution domain because it is capable of modeling quite arbitrary and irregular geometries and has long been used to solve problems with cornplex geometries vay successfdy. No global topology or or thogonalit y res trietions is required for finite-dement grid. Roundaries are automatically fitted and there is no restriction in concentrating the grid. Besides the finite-element and finite-clifFerence methods, there are the contirol-volume approaches which have the advantage of providing a conservative discretization of the governing equations. Those approaches allow exact numerical conservation of the conserved quantities in each finite control volume.

A.2 Finite-Element Discret ization

In the finite-element method, the solution domain is broken into s number of sub- regions which are cded elements, Fig.A. 1. The vertices of the element are nodal locations of the domain. They are the location of the problem unknowns. Ditferent element shapes are used in fiaite-element method. Regarding the simplicity and economy of the solution, quadrilateral elements have ben selected in this study.

In Fig.A.2, a single element is separated fkom the 0th- elements in order to expand the fiaite-element relations for it. As seen, there are hocoordinate systems, APPENDIX A. ELEMENT GEOMETlUC RELATIONS a global orthogonal eoordinate system of (z,y) and a local non-orthogonal coordinate system of (t,~).The ranges of t and r) are nom -1 to +l within the dement. This local co

Figure A. 1: Finite-element discretization within a nozzle domain.

Figure A.2: An isolated element. APPENDIX A. ELEMENT GEOMETRIC RELATIONS 226 local system, it is necessary to relate local and global coordinates. Fite-element shape functions, Ni with i=l ...4, are used to connect th-, i.e.

where xi and y6 are the coordinates of node i. In this study, ne use bilinear shape- functions which are defined by

A.3 Local-Global Coordinate Tkansformation

The procedure of the discretization of the governing equations requires sewral dif- ferentiations and integrations to be performed within an element. Simple connection presented in Eq.(A.2) does not provide enough device to do that. In order to develop our local-global transformations ne, start with differentiation of the dependent variable 4 as APPENDLX A. ELEMENT GEOMETRIC RELATIONS 227

Since and are not known the hain deis used to convert them to suitable for m

Here all terms are hown except and 3 terms. Thus, we transfom them into the following matrir structure BBZi @% % (a.)-G = [$ g] (a".)* the solution of which is

where J is the Jacobian of transformation and is d&ed by

Equation (A.6) is used to derive and forms as a function of and forms in the foliowing manner. Initially, the foIIowing differentid forms are obtained fiom Eq.(A.l) as APPENDIX A. ELEMENT GEOMETRIC RELATIONS 228

Next ,shape hiaction derivatives with respect to C and q are derived from Eq. (A.2),

This cornpletes the process of eomputing and forms.

In the next step, we pay attention to the process of integration oves an arbitrary sub-domain ( = O + tl and r) = O -t m. If the vector T is dehed as

(A. 10) its differential form is given by

The area or the volume per unit depth of this sub-domain is calculated by the produet of df and df, i.e.,

m=~&xdrj~ (A.12)

(A.13)

Considering Eq. (A.?), this equation is simplified to

(A.14)

The integration over the defined domain nill yield

(A. 15) Appendix B

Density Linearizat ion

Since density is not a major dependent variable it is necessary to comect it to the other major dependent variables. This variable appears in many terms of the governing equations and their discretized forms, e.g . the transient terms. The density in incompressible flow is constant and needs no treatment. There is no difficulty for computing the density if it is a lagged value. However, the density needs treatment whenever it appears as an unknown in our equations in compressible flow. The ap propriate conneetion is provided by the equation of state and specifically the ideal gas law

As seen, the density is a function of both pressure and temperature. A simple linearization for this equation is provided by considering an active role for P and a lagged for T as This form of lineafization is excellent for the compressible flows with isothermal assumptions. The overline denotes a lagged due fiom the previons inner/outer iterations* Anothes method of treatment is to exnploy a Taylor series expansion which considers active roles for both P and T. It is given by

The density differential forms are derived fiom &.(BA), i.e.,

Now, these expressions are substituted in Eq.(B.3) and the results are simplified to

This linearization is similar to a Newton-Raphson linearization, Anderson et al [6Z]. In this form of hearization, s nonlinear product of two parameters of A and B is linearized to

AB= BA+AB-AB

This form of linearization is cailed many times during this thesis. Appendix C

Linearization of Moment um Convections

Before presenting the different possible linearizations for the nonlinear convection terms of the momentum equations, it is necessary to have an introduction to the concept of the individual elements in those terms. These tams need to be studied more carefully than the other terms because two diffetent concepts could be defined for t hose individuals.

To expand those concepts, it is helpfd to recall the method of derivation of those tmsin the momentum equations. White [57] presents the basic dinerentid equations by considering an elemental control volume, Figure C. 1. The balance of the mass and the convection part of the linear-momentnm conservation equations could be written for this infinitesimai fixed control volume as

i but i lin C. LRVEARIZATION OF MOMENTUM CONVECTIONS 232

Figure C.l: A control volume showing inlet and outlet mass flows on the x faces. where dz and dy present the control volume dimensions and S indicates the area of each control surface. Table C.l presents the mass and momentun fluxes thrmgh the control volume faces. If these terms are plugged in Eq. (C. 1) and Eq. (C.2) the results after some simplifications are given by

As is seen, the velocity components appear in two positions in each term of Eq. (C.4),either as a part of the mass flux components or as a veloaty components. This causes two difkent meanings for these two velocities. The velocity compe nents which appear in the mass flux components are named conuecting ueloeities or mas cowerving velocities. This name retums to Eq.(C.S) where these velocity components numerically satisfy the mass conservation eguation. They convect Faces Idet mass flux Outlet mass flux

2 pu dv lptr+&pu)~ldv , Y pu& [P"+&(pv)d4& ,, Faces Inlet momenttun flux Outlet momentum flux L 2 ptiy ~puY+$(PUY) dz] dg Y pvy dx $(&) dy] F [prq + Table C. 1: Mass and linear momentum flwces for control volume in Figure C. 1. through the control volume. On the other hand, the components of ? in Eq.(C.4) are called conuected uelocities. These velocities are convected by the mass fluxes through the control volume surfaces. In order to distinguish these two types of velocities, The convecting is identified by a hat (^). Now, we can expand Eq.(C.4) in z and y directions while th& original concepts are retained

x - direction

y - direction

It is worthf'ul to note that an arbitrary switch from convected to convecting or from convecting to convected may destnict the original concepts which derivation of the equations are based on.

Generally spealing, we study two different forms of the linearization for the convection terms of the momentum equations. At this stage, we are concerned with the equations which are derived based on control volume discretkation. Table 2.1 shows the result of simple linearization of the convection terms of the momeentum equations which are repeated here with considering the above concepts - pu u =û(p16) - pû u O(p) - pû v =5U(p) - pV u nû(pu)

A second form of linearization is possible using a Newton-Raphson linearization de, Eq.(B.6). This linearization considers more active roles for the individuals in the nonlinear term. We show the procedure for the linearization of the pu term which is linearized in the foliowing manner

Since 4 is not a major unknown in this study, one more Newton-Raphson linearization is used for u = 7,i.e.,

If Eq. (C.8) is plugged in Eq. (C.7) and some more simplifications are done the result is

This linearization results in active roles for both convected and convecting velocities as weil as density in x-momentam equation term. The density tam is simply corn- puted by the lagged dues fiom the previous iterations. Considering this approximation and the dehition of momentum components, ne define a general equation APPENDIX C. LnVEAIUZATION OF MOlMENTUM CONVECTIONS 235

in order to include both foam of the simple and Newton-Raphscm linearizations, i.e. Eq.(C.6) and Eq.(C.9),

where k' =O results in simple linearization and kf=l represent Newton-Raphson linearization. This procedure codd be aimilarly followed for the tkee other convection terms in the momentum equations. The results will be

(C. 12)

(C. 13)

Eq. (C. IO) was tes ted in the onedimensional investigation with success. It generaüy showed bet ter results than the simple linearization.

One more form of linearization was tested in the oneaimensional investigations. In this form the two concepts of convected and convecting were mixed up and the following linearization was derived,

(C. 14)

where k' and k" are constants which make the two linearizations possible. The consideration of k'=kt'=l results in Newton-Raphson linearization and the consideration of kg=$ and kf'=O results in a simple linearization form. This form of Iùiearization generaüy showed fater convergence than the previous ones.

The discussion in this appendix is important in colocated grid fodation where the decouphg of pressure and velocity fieids may happen. Appendix D

Linearizat ion of Moment um Diffusion Terms

The terms involving the differential form of velocity components, i.e. e, g,2, and -au terms, are nonlinear if the momentum components are selected as the dependent variables. There are methods t O linearize t hese nonlinearities. Here, we present the result of linearization only for 2 with this knowledge that the other three foms are linearized in the same manner. The first method uses the advantages of the chin rule to derive an appropriate expression for $ as

Rearrangement of this expression yields

The nonlinear density variable, which is derived by the spatial discretization of the second term in the right-hand-side of the eqoation, could either be iinearized using the methods presented in Appendix B or be lagged from the known values of the APPENDIX D. LLNEARIZATION OF MOMENTUM DIFFUSION TERJUS 237 previous iterations. The latter has been chosen to treat the above density term t hroughout this t hesis.

In the second method, the second term in right-hand-side of Eq.(D.2) is replaced

The next step is to linearize this term as

This form of linearization was not used in this thesis. Appendix E

Convection in Non-conservat ive Momentum

The following investigation was performed for t wo terms of the one-dimensional moment- equation, i.e. pu& and uy.This investigation presents the possible weak forms of discretization for the above tenns. One rnethod for linearizing pu$ tenn the momentum equation is the use of lagged values for g. This results in a direct transfer of this term to left hand side of the integration point equation. This form of linearization is studied for the steady-state Euler flow which is derived fiom Eq.(3.21) and linearized as

A central difference for d result in

The investigation for a constant pressure field shows that this equation results in a wrong approximation for the integration point due. APPENDLX E. CONVECTION IN NON-CONSERVATIVEMOMENTUM 239

In the nad step, an apwind clifference is considered for 2 in Eq. (E. 1). Now, if - -82 is approximated by an upaind ciifference, the resdt is written as

- and a central clifference for $ wiIl result in

Similarly, the investigation for a constant pressure field shows that these two approximations are not appropriate approximations do resdt in wrong evahations of the integration point values. Indeed, both of them are subject to negative or infinite coeflicient s for F, .

Similarly, Equation (3.21) is recalled with the assnmption of the steady-state inviscid flow in order to expand the investigation for u& tem of that equation,

A central clifference for and an upwind difference for $ dlfinally result in

This expression also presents a wrong approximation for the integration point value. The correct approximation for the velocity at the integration point would be the average of the neighboring nodal velocity values if the pressure field is constant. Appendix F

Linearization of Energy Transient Term

Tliere are a number of techniques to deal with the nonlinear transient term of the energy equation in conservative treatment. In all cases the original form of this term is discretized by mass lumped approach and using backward scheme in the,

Here, the concern is on the nonlinear form of the first tam inside the parenthesis. A simple way to linearize this term is to substitute E fiom Eq.(2.13) and try to linearize the equation appropriately as

The other form of linearization is derived by the use of the Newton-Raphson linearization scheme, Eq. (B .6), APPENDLX F. LIM3ARIZATION OF ENERGY TRANSIENT TERM 241

In this form, e is substituted fkom Eq.(B.5) and E is treated as

After the substitution of these nonlinear treatments in Eq. (F.3)and some rearrangement, the foilowing form is resulted

This form of lindzation produces negative coetncient for temperature and cannot be reliable.

Another form of linearization is possible if Eq.(F.l) is treated in another form as

Similar to Eq.(F.3), nonlinear form of eT is changed to eT + Fe - and e is linearized by Eq.(B.à) and is substituted in it. The final result is given by

--P+-F+-G--a(pe) G Ü V e0EO dû RAB 2A8 2A8 AB which does not include T term. This form was not used in this work. Appendix G

Linearization of Energy Convection Terms

Regarding the conservative treatment of the convection terms in the enzrgr equation, a number of approaches is applicable. The two convection terms can be mitten either in the enthalpy form 9+ Sv or in the energy form + B.eV Here we are concerned ody on enthalpy form and restnct the approaches to two possible form of linearizations. In this study, the original concepts of mvected and convecting, which were described in Appendir C, are respected. For example,

The fist linearization approach uses the definition of h, Eq.(2.14), and substitutes it directly in convection term. The remainder is to linearize the resuited expressions appropriately respect to the dependent variables, APPENDLX G. LINEARIZATION OF ENERGY CONVECTION TERMS 243

If the concept of convected and convecting is not respected these equations are written as

A second linearization uses the Newton-Raphson schema, i.e. Eq. (B.6), to linearize these convection terms. It is given by

Using the definition of enthalpy, substitutiag it in above equations, and treating them in an appropriate manner wodd result in

Ignoring the concepts of convect ing and convect ed result s in

Ali lagged values of these equations can be calculated fiom the previous iterations by using the known values of eitha conveeted or conveeüng variables. Appendix H

Streamwise Discretization Approach

In two-dimensional flows, the direction of the flow has an important role in discretizing some differential tmsof the governing equations. In a one-dimensional flow, grid lines and flow direction are coincident, however, this is not the case for the twdimensional flows. The importance of the flow direction is in evaluating some diff'ential terms in streamwise direction. The inconsistency between grid lines and flow directions does not allow discretization in streamwise direction. For tunately, convection Merential terms which are sensible to flow direction could be combined and written in streamwise direction as

These terms can be writ ten in the local streamwise direction as APPENDIX H. STREAMWfSE DISCl?.ETIZATION APPROACH where

and

Eq. (H.2)provides a straightfomard difkrential form to be used for discretization in streamwise direction like streamaOse upwinding, central-diff'erencing, etc. Appendix 1

Laplacian Operator Discretization

The procedure of computing integration point expressions requires the discretization of the diffusion terms which may take Laplaeian foxm. Finite element discretization of the nonlinear Laplacian form is not straightforward. Here, the ap proach was taken by Schneider [71] is followed. In this regard, consider the Lapla- uan of an scdar 4 which is given by

Considering Fig.I.1, we can obtain some approximations for terms of Eq.(I.l)at integration point of 1 as

If these expressions are substituted in Eq.(I.l) the following enpression be resdted atta some remangement APPENDIX 1. LAPLACIAN OPERATOR DISCRETIZATION

Figure 1.1: Laplacian length scale.

Which the diffnsion length scale is defined as

As is observed, the derived approximation for the Laplacian Eq.(I.3) satisfact* rily satisfy the limiting case of diffusion dominated flows. In another words, when

Eq.(I.3) is reduced to the bilinear interpolation form of the nodal values, i.e.,

Which is a correct approximation.

Eq.(I.3) was derived based on a rectangular element shape. However, it does not result in a correct approximation for gendfom of the quadrilateral finite eiements which we use during this study. For a general fotm of elementr, ne use APPENDLX 1. LAPLACLAN OPERATOR DISCRETIZATION 248

Eq.(I.3) with a corrected Ld. hrthermore, Az and Ag are replaced by the length scalea perpendicalar and tangentid, respectively, to the face in question. This means Ag is considered as the length of the face in question and Az is determined b~9

where 1JI is the magnitude of the Jacobian of transformation, Eq.(A.?).